Reflection Positive Stochastic Processes Indexed by Lie Groups
Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symme...
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irk-123456789-1477542019-02-16T01:24:20Z Reflection Positive Stochastic Processes Indexed by Lie Groups Jorgensen, P.E.T. Neeb, K.H. Ólafsson, G. Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations. 2016 Article Reflection Positive Stochastic Processes Indexed by Lie Groups / P.E.T. Jorgensen, K.H. Neeb, G. Ólafsson // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 68 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 22E45; 60G15; 81S40 DOI:10.3842/SIGMA.2016.058 http://dspace.nbuv.gov.ua/handle/123456789/147754 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations. |
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Article |
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Jorgensen, P.E.T. Neeb, K.H. Ólafsson, G. |
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Jorgensen, P.E.T. Neeb, K.H. Ólafsson, G. Reflection Positive Stochastic Processes Indexed by Lie Groups Symmetry, Integrability and Geometry: Methods and Applications |
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Jorgensen, P.E.T. Neeb, K.H. Ólafsson, G. |
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Jorgensen, P.E.T. |
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Reflection Positive Stochastic Processes Indexed by Lie Groups |
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Reflection Positive Stochastic Processes Indexed by Lie Groups |
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Reflection Positive Stochastic Processes Indexed by Lie Groups |
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Reflection Positive Stochastic Processes Indexed by Lie Groups |
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Reflection Positive Stochastic Processes Indexed by Lie Groups |
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reflection positive stochastic processes indexed by lie groups |
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Інститут математики НАН України |
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2016 |
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Reflection Positive Stochastic Processes Indexed by Lie Groups / P.E.T. Jorgensen, K.H. Neeb, G. Ólafsson // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 68 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT jorgensenpet reflectionpositivestochasticprocessesindexedbyliegroups AT neebkh reflectionpositivestochasticprocessesindexedbyliegroups AT olafssong reflectionpositivestochasticprocessesindexedbyliegroups |
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2025-07-11T02:46:30Z |
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2025-07-11T02:46:30Z |
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1837317010095079424 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 058, 49 pages
Reflection Positive Stochastic Processes
Indexed by Lie Groups
Palle E.T. JORGENSEN †, Karl-Hermann NEEB ‡ and Gestur ÓLAFSSON §
† Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
E-mail: palle-jorgensen@uiowa.edu
‡ Department Mathematik, FAU Erlangen-Nürnberg,
Cauerstrasse 11, 91058-Erlangen, Germany
E-mail: neeb@math.fau.de
§ Department of mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
E-mail: olafsson@math.lsu.edu
Received October 28, 2015, in final form June 09, 2016; Published online June 21, 2016
http://dx.doi.org/10.3842/SIGMA.2016.058
Abstract. Reflection positivity originates from one of the Osterwalder–Schrader axioms for
constructive quantum field theory. It serves as a bridge between euclidean and relativistic
quantum field theory. In mathematics, more specifically, in representation theory, it is
related to the Cartan duality of symmetric Lie groups (Lie groups with an involution)
and results in a transformation of a unitary representation of a symmetric Lie group to
a unitary representation of its Cartan dual. In this article we continue our investigation
of representation theoretic aspects of reflection positivity by discussing reflection positive
Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian
measures in spaces of distribution vectors. This provides new constructions of reflection
positive unitary representations.
Key words: reflection positivity; stochastic process; unitary representations
2010 Mathematics Subject Classification: 22E45; 60G15; 81S40
Contents
1 Introduction 2
2 Reflection positive representations 4
3 Reflection positive Lie group actions on measure spaces 7
3.1 (G,S, τ)-measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Stochastic processes indexed by Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Associated positive semigroup structures and reconstruction . . . . . . . . . . . . . . . . . 15
3.4 Klein’s characterization of the Markov case . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Total subsemigroups of Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 The discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Measures on path spaces for topological groups 25
4.1 One-parameter convolution semigroups of measures on polish groups . . . . . . . . . . . . 25
4.2 Standard path space structures for locally compact groups . . . . . . . . . . . . . . . . . . 29
5 Gaussian (G,S, τ)-probability spaces 31
5.1 Second quantization and gaussian processes . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Application to reflection positive representations . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Equivalence of gaussians measures for reproducing kernel Hilbert spaces . . . . . . . . . . 34
mailto:palle-jorgensen@uiowa.edu
mailto:neeb@math.fau.de
mailto:olafsson@math.lsu.edu
http://dx.doi.org/10.3842/SIGMA.2016.058
2 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
5.4 Gaussian measures on distributions on Lie groups . . . . . . . . . . . . . . . . . . . . . . . 36
5.5 Ergodicity of γH for G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A Continuity of a stochastic process on G 43
B Markov kernels 44
References 47
1 Introduction
Reflection positivity is one of the cornerstones in constructive quantum field theory. It was
first formulated in the fundamental work of Osterwalder and Schrader [57, 58], see also [20].
Mathematically, a quantum physical system corresponds to a unitary representation of the
corresponding symmetry group G. In euclidean quantum field theory this group is the euclidean
motion group and in relativistic quantum field theory the symmetry group is the Poincaré group.
Reflection positivity enters the picture when it comes to passing from euclidean quantum fields
to relativistic ones. The time reflection and the passage to imaginary time, sometimes called
Wick rotation, changes the standard euclidean inner product into the Lorentz form of relativity.
The time reflection corresponds to an involution τ of the euclidean motion group G and then
the duality of symmetric Lie algebras g = gτ ⊕ g−τ ↔ gc = gτ ⊕ ig−τ leads to a duality between
the Lie algebra of the euclidean motion group and that of the Poincaré group.
In the following we call a pair (G, τ), consisting of a Lie group G and an involutive auto-
morphism τ : G → G a symmetric Lie group. On the group level, the aforementioned duality
is implemented by the Cartan duality (or c-duality for short) G ↔ Gc, where Gc is the simply
connected Lie group with Lie algebra gc. Passing from real to imaginary time and from the
euclidean motion group to the universal covering of the (identity component of the) Poincaré
group is an important special case of c-duality which has been studied independently from the
quantum field theoretic context.
In addition to c-duality, the other three basic notions considered in the basic theory of
reflection positivity are that of reflection positive Hilbert spaces, reflection positive kernels,
and reflection positive representations. A reflection positive representation together with the
Osterwalder–Schrader quantization leads to an infinitesimally unitary representation of the Lie
algebra gc. The problem in representation theory is then to determine if this representation arises
as the derived representation of a unitary representation of Gc, which establishes a passage from
a unitary representations of G to one of Gc.
This integration process is often accomplished using reflection positive kernels and geometric
actions of the Lie group or its Lie algebra [46] or, as in the case of the Lüscher–Mack theorem [45],
using semigroups and invariant cones. A second step is then to determine the resulting repre-
sentation in terms of decomposition into irreducible representation. For further representation
theoretic results related to reflection positivity, we refer to [26, 33, 34, 35, 36, 49, 50, 51, 56, 60].
Our present paper concentrates on various mathematical aspects of constructions of reflection
positive representations: infinite-dimensional analysis, functional integration and gaussian mea-
sures, and stochastic processes. On the way we recall several basic facts in these areas to make
this article more self-contained.
Shortly after the groundbreaking work of Osterwalder and Schrader, A. Klein and L. Landau
built a bridge between reflection positivity, path spaces and stochastic processes [40, 41, 42].
One of our goals in this article is to connect the ideas of Klein and Landau to representation
theory and our previous work on reflection positive representations [35, 36, 49, 50, 51]. This is
done by replacing the real line, viewed as continuous time, by an arbitrary Lie group G, not
necessarily finite-dimensional, and the positive time axes R+ by a semigroup S ⊂ G invariant
Reflection Positive Stochastic Processes Indexed by Lie Groups 3
under s 7→ s# = τ(s)−1. This leads naturally to the concepts of (G, τ)-measure spaces, reflection
positive measure spaces, and a positive semigroup structures introduced in Section 3 of this
article. Many concepts extend naturally to general triples (G,S, τ). But in the generalization of
the Abel–Klein reconstruction theorem (Theorem 3.22) which reconstructs a (G,S, τ)-measure
space from a positive semigroup structure, we need to assume that G = S ∪ S−1; then S is
called total. It would be interesting to see how far our techniques can be extended beyond total
subsemigroups.
In the following section we turn back to the classical case where the symmetric Lie group
is (R,R+,− id), but the measure space is the path space P (Q) = QR of a polish topological
group Q, mostly assumed locally compact. If ν is a measure on Q and (µt)t≥0 is a convolution
semigroup of symmetric probability measures on Q satisfying ν ∗ µt = ν for every t > 0, then
the reconstruction theorem applies to the corresponding R+-action on L2(Q, ν) and leads to
an invariant probability measure on the path space QR corresponding to a reflection positive
representation (Theorem 4.9). However, the corresponding measure on P (Q) is the product of
the measure ν, on the constant paths, and a probability measure on the pinned paths
P∗(Q) := {ω ∈ P (Q) : ω(0) = 1},
which is also invariant under a suitable one-parameter group of transformations. Since we
presently do not know how to obtain a similar factorization if (R,R+,− idR) is replaced by some
(G,S, τ), we discuss the one-dimensional case in some detail.
One of the advantages of the reflection positivity condition is that it allows to construct
representations of a symmetric Lie group (classically the euclidean motion group) in L2-spaces
of measures on spaces of distributions, which after the Osterwalder–Schrader quantization lead to
unitary representations of the c-dual group. In this connection the physics literature considered
reflection positive distributions D ∈ S ′, invariant under the euclidean group. The Hilbert
space HD ⊆ S ′ specified by the corresponding positive definite kernel [61] leads to a reflection
positive representation of the euclidean motion group [49, 50]. This motivates our discussion
of G-invariant gaussian measures corresponding to a unitary representation (π,H) of G. The
replacement for the Gelfand triple (S, L2(Rn),S ′) is the Gelfand triple (H∞,H,H−∞), whereH∞
denotes the Fréchet space of smooth vectors in H and H−∞ is it conjugate linear dual (the space
of distribution vectors). Here the natural question is for which representations the gaussian
measure ofH can be realized onH−∞? A partial answer is given in Theorem 5.17, Corollary 5.19,
and Proposition 5.20. All of this links naturally to the theory of generalized Wiener spaces and
white noise processes for Rn, indexed by L2(Rn); see, e.g., [2, 3, 5, 29].
Finally we would like to mention that there is also an important branch of applications of
reflection positivity in statistical physics which does not refer to semigroups at all; see [19] and
the more recent [51], where the corresponding group G may be a torus, hence does not contain
any proper open subsemigroup. We hope to develop this point in future work.
Our article is organized as follows. In the first section we start by recalling the basic facts
about reflection positive Hilbert spaces and reflection positive representations [49, 50]. We
always have a symmetric Lie group (G, τ) and a subsemigroup S invariant under s# = τ(s)−1.
Based on ideas from [42], we introduce the special class of reflection positive Hilbert spaces of
Markov type. The main result is Proposition 2.9.
In Section 3 we discuss stochastic processes indexed by a symmetric Lie group (G, τ), where
the forward time R+ is replaced by a subsemigroup S ⊂ G invariant under s 7→ s] = τ(s)−1. Here
we prove our generalization of the reconstruction theorem (Theorem 3.22) which reconstructs for
a so-called positive semigroup structure a (G,S, τ)-measure space from which it can be derived.
From the representation theoretic perspective, it corresponds to finding euclidean realizations
of unitary representations of the c-dual group Gc. Unfortunately, the reconstruction process
requires that G = S ∪ S−1, a property which is briefly discussed in Section 3.5.
4 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
In Section 4 we build a bridge between Markov processes and the reconstruction process. To
this end, we return to the classical setting, where the symmetric Lie group is (R,R+,− id), but
the measure space is the path space P (Q) = QR of a polish group Q and the corresponding
(R,R+,− idR)-measure space is given by a one-parameter group Ptf = f ∗ µt, t ≥ 0, of left
invariant Markov kernels on some space L2(Q, ν). Here our main result concerns a factorization
of the measure space P (Q) as Q × P∗(Q) and a corresponding factorization of the measure
preserving action of R. For the special case where µt is the gaussian semigroup on Rd, the
corresponding measure on P∗(Rd) is the Wiener measure, but the measure on P (Rd) is a product
of Lebesgue and Wiener measure. The main results in Section 4 is Theorem 4.9 which relates
all this to the reconstruction process and hence to reflection positive representations.
In Section 5 we discuss the second quantization functor and how it can be used to derive
from an orthogonal reflection positive representation a gaussian (G,S, τ)-probability spaces
(Proposition 5.6). To understand the ambiguity in this construction we also discuss equivalence
of gaussian measures for reproducing kernel Hilbert spaces, and in Subsection 5.4 we connect
this issue to our previous work [49] on distributions on G and distribution vectors of a unitary
representation of G. In Theorem 5.17 we determine when a gaussian measure γH of a Hilbert
space H can be realized in the dual of the space
⋂
D(An) of smooth vector of a self adjoint
operator A. This is then applied to a unitary representation of G in Corollary 5.19.
The article ends with two appendixes where needed material on stochastic processes and
Markov semigroups is collected.
2 Reflection positive representations
In this preliminary section we collect some results and definitions from [49, 50] concerning
reflection positive representations.
Definition 2.1. A reflection positive Hilbert space is a triple (E , E+, θ), where E is a Hilbert
space, θ a unitary involution and E+ a closed subspace which is θ-positive in the sense that the
hermitian form 〈v, w〉θ := 〈θv, w〉 is positive semidefinite on E+.
For a reflection positive Hilbert space (E , E+, θ), let N := {u ∈ E+ : 〈θu, u〉 = 0} and let Ê be
the completion of E+/N with respect to the inner product 〈 · , · 〉θ. Let q : E+ → Ê , v 7→ q(v) = v̂
be the canonical map. Then Eθ+ := {v ∈ E+ : θv = v} is the maximal subspace of E+ on which q
is isometric.
Definition 2.2. Let (E , E+, θ) be a reflection positive Hilbert space. If E0 ⊆ Eθ+ is a closed
subspace, E− := θ(E+), and E0, E± the orthogonal projections onto E0 and E±, then we say that
(E , E0, E+, θ) is a reflection positive Hilbert space of Markov type if
E+E0E− = E+E−. (2.1)
Lemma 2.3. Suppose that E is a Hilbert space, θ : E → E is a unitary involution, E+ is a closed
subspace and E0 ⊆ Eθ+. Let E− = θ(E+).
(i) If (E , E0, E+, θ) satisfies the Markov condition (2.1), then E+ is θ-positive, E0 = Eθ+, N =
E+ E0 and q : E0 → Ê is a unitary isomorphism.
(ii) If q0 : E0 → Ê is a unitary isomorphism, then (2.1) holds.
Proof. (i) By definition, we have θE+ = E−θ so Eθ+ ⊆ E+ ∩ E−. Let u ∈ Eθ+ E0. Then the
operator on the right hand side of (2.1) reproduces u while the left hand side yields 0. Hence
u = 0 and E0 = Eθ+. As E0 ⊂ Eθ+ it follows that E0θ = θE0 = E0 which implies that
E+θE+ = E+E−θ = E+E0E−θ = E0θE+ = E0E+ = E0.
Reflection Positive Stochastic Processes Indexed by Lie Groups 5
It follows that, for u ∈ E+, we have
〈θu, u〉 = 〈E+θE+u, u〉 = 〈E0u, u〉 = ‖E0u‖2 ≥ 0.
We obtain in particular that ‖q(v)‖ = ‖E0v‖ for v ∈ E+, so that q|E0 : E0 → Ê is a unitary
isomorphism.
(ii) Since q|Eθ+ is isometric, we obtain E0 = Eθ+ and thus N = E+ ∩ θ(E0)⊥ = E+ E0. This
leads to the orthogonal decomposition E+ +E− = θ(N )⊕E0⊕N and to E0 = E+∩E−. Now (2.1)
follows. �
In the following, if (E , E0, E+, θ) satisfies the Markov condition (2.1), then we will always
assume that E0 = Eθ+ and hence only write (E , E+, θ).
Definition 2.4. If τ is an involutive automorphism of the Lie group G, then we call (G, τ)
a symmetric Lie group. A symmetric semigroup is a triple (G,S, τ), where (G, τ) is a symmetric
Lie group and S ⊆ G is a subsemigroup satisfying
(S1) S is invariant under s 7→ s] = τ(s)−1,
(S2) HS = S for H := (Gτ )0,
(S3) 1 ∈ S.
We define a left invariant partial order ≺S on G by g ≺S h if g−1h ∈ S, i.e., h ∈ gS.
Example 2.5. (1) (R,R+,− idR) and (Z,N0,− idZ) are the most elementary examples of sym-
metric semigroups.
(2) Semigroups with polar decomposition: Let (G, τ) be a symmetric Lie group and H be an
open subgroup of Gτ := {g ∈ G : τ(g) = g}. We denote the derived involution g → g by the
same letter and define h = {x ∈ g : τ(x) = x} = gτ and q = {x ∈ g : τ(x) = −x} = g−τ . Then
g = h⊕ q. We say that the open subsemigroup S ⊆ G has a polar decomposition if there exists
an H-invariant open convex cone C ⊂ q containing no affine lines such that S = H expC and
the map H×C → S, h,X 7→ h expX is a diffeomorphism (cf. [26, 44, 48]). Typical examples are
the complex Olshanski semigroups in complex simple Lie groups such as SUp,q(C)C ∼= SLp+q(C).
They exist if and only if G/K is a bounded symmetric domain. This is equivalent to the existence
of a G-invariant convex cone C ⊂ ig such that G expC is a semigroup. More generally we have
the causal symmetric spaces of non-compact type like SOo(1, n + 1)/ SOo(1, n). In this case
q ' Rn+1 and C can be taken as the open light-cone.
(3) The simply connected covering group G := S̃L2(R) of SL2(R) carries an involution τ
acting on sl2(R) by
τ
(
x y
z −x
)
=
(
x −y
−z −x
)
,
and there exists a closed subsemigroup S ⊆ G whose boundary is
∂S = H(S) := S ∩ S−1 = exp(b) with b :=
{(
x y
0 −x
)
: x, y ∈ R
}
.
This semigroup satisfies S] = S, the subgroup H(S) is τ -invariant, but strictly larger than Gτ0
(see also Section 3.5 for more on this semigroup).
Definition 2.6. For a symmetric semigroup (G,S, τ), a unitary representation U of G on
(E , E+, θ) is called reflection positive if θUgθ = Uτ(g) for g ∈ G and UsE+ ⊆ E+ for every s ∈ S.
6 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Remark 2.7. Let Gτ = Go {1, τ}. Then θUgθ = Uτ(g) holds for every g ∈ G if and only if U
extends to a unitary representation of Gτ by defining Uτ = θ.
Remark 2.8. (a) If (Ug)g∈G is a reflection positive representation of (G,S, τ) on (E , E+, θ),
then we obtain contractions (Ûs)s∈S , on Ê , determined by
Ûs ◦ q = q ◦ Us|E+ ,
and this leads to an involutive representation (Û , Ê) of S by contractions (cf. [35, Corollary 3.2]
and [49]). We then call (U, E , E+, θ) a euclidean realization of (Û , Ê). We refer to [46] and [45]
for methods for obtaining a unitary representation of Gc from Û .
(b) For (G,S, τ) = (R,R+,− idR), continuous reflection positive unitary one-parameter
groups (Ut)t∈R lead to a strongly continuous semigroup (Û , Ê) of hermitian contractions and
every such semigroup (C,H) has a natural euclidean realization obtained as the GNS repre-
sentation associated to the positive definite operator-valued function ϕ(t) := C|t|, t ∈ R [50,
Proposition 6.1].
The following proposition is a generalization of [50, Proposition 5.17] which applies to the
special case (R,R+,− idR).
Proposition 2.9. Let (Ug)g∈G be a reflection positive unitary representation of (G,S, τ) on
(E , E+, θ), let E0 ⊆ (E+)θ be a subspace and Γ = q|E0 : E0 → Ê. If (E , E0, E+, θ) is of Markov type,
then the following assertions hold:
(i) The reflection positive function ϕ : G→ B(E0), ϕ(g) := E0UgE0, is multiplicative on S.
(ii) ϕ(s) = Γ∗ÛsΓ for s ∈ S, i.e., Γ intertwines ϕ|S with Û .
Proof. (i) Let K ⊆ H be the U -invariant closed subspace generated by E0. Let (E0)G denote
the linear space of all maps G→ E0. Then the map
Φ: K → (E0)G, Φ(v)(g) := E0Ugv
is an equivalence of the representation U of G on K with the GNS representation defined by ϕ,
and the representation Û of S on Ê is equivalent to the GNS representation defined by ϕ|S ,
where the map q : E+ → Ê corresponds to the restriction f 7→ f |S [49, Proposition 1.11]. The
inclusion ι : E0 ↪→ Hϕ is given by ι(v)(g) = E0Ugv = ϕ(g)v for g ∈ G, and likewise the inclusion
ι̂ : E0 ↪→ Hϕ|S is given by ι(v)(s) = ϕ(s)v for s ∈ S. Lemma 2.3 implies the surjectivity of the
inclusion ι̂. In view of [50, Lemma 10.3], this is equivalent to the multiplicativity of ϕ|S .
(ii) If Γ is unitary, then [50, Lemma 5.16(ii)] implies that q = Γ ◦ E0|E+ . For s ∈ S, the
relation Ûs ◦ q = q ◦Us|E+ leads to ÛsΓE0|E+ = ΓE0Us|E+ , so that Γ∗ÛsΓ = E0UsE0 = ϕ(s), i.e.,
Γ intertwines ϕ(s) with Ûs. �
Lemma 2.10. Assume that U is a reflection positive representation of (G,S, τ) on (E , E+, θ).
Let E0 ⊆ Eθ+. If ϕ(g) = E0UgE0 is multiplicative and E0 is S-cyclic in E+ then (E , E+, θ) is of
Markov type.
Proof. As in the proof of Lemma 2.3, this follows from [50, Lemma 10.3]. �
Remark 2.11. If G = S ∪ S−1 and E0 is G-cyclic in E . Then E0 is S-cyclic in E+ and E+ + E−
is dense in E , so that the Markov condition leads to E = N ⊕ E0 ⊕ θ(N ).
Reflection Positive Stochastic Processes Indexed by Lie Groups 7
3 Reflection positive Lie group actions on measure spaces
In this section we generalize several results from [41] and [42] to situations where (R,R+,− idR)
is replaced by a symmetric semigroup (G,S, τ). This leads us to the concept of a (G,S, τ)-
measure space generalizing Klein’s Osterwalder–Schrader path spaces for (R,R+,− idR). The
Markov (G,S, τ)-measure spaces generalize the path spaces studied by Klein and Landau in [42].
The main result of this section is the correspondence between (G,S, τ)-measure spaces and
the corresponding positive semigroup structures. For (G,S, τ) = (R,R+,− idR) this has been
developed in [41, 42], motivated by Nelson’s work on the Feynman–Kac formula [54].
3.1 (G,S, τ )-measure spaces
In this section we introduce basic concepts related to (G,S, τ)-measure spaces and draw some
consequences from the definitions.
Definition 3.1. Let (G, τ) be a group with an involutive automorphism τ . A (G, τ)-measure
space is a quadruple ((Q,Σ, µ),Σ0, U, θ) consisting of the following ingredients:
(GP1) a measure space (Q,Σ, µ),
(GP2) a sub-σ-algebra Σ0 of Σ,
(GP3) a measure preserving action U : G→ Aut(A) on the W ∗-algebra A := L∞(Q,Σ, µ) which
is strongly continuous in measure, i.e.1,
lim
g→1
µ(|Ugf − f | ≥ ε) = 0 for ε > 0 and f ∈ A ∩ L1(Q,Σ, µ),
and
(GP4) a measure preserving involutive automorphism θ of L∞(Q,Σ, µ) for which θUgθ = Uτ(g)
for g ∈ G and θE0θ = E0, where E0 : L∞(Q,Σ, µ) → L∞(Q,Σ0, µ) is the conditional
expectation.
(GP5) Σ is generated by the sub-σ-algebras Σg := UgΣ0, g ∈ G.
If µ is a probability measure, we speak of a (G, τ)-probability space. If (G,S, τ) is a symmetric
semigroup, then we write Σ± for the sub-σ-algebra generated by (Σs)s∈S±1 , and E± for the
corresponding conditional expectations.
Definition 3.2. (a) A (G, τ)-measure space is called reflection positive with respect to S if
〈θf, f〉 ≥ 0 for f ∈ E+ := L2(Q,Σ+, µ).
This is equivalent to E+θE+ ≥ 0 as an operator on L2(Q,Σ, µ) and obviously implies θE0 = E0.
If this condition is satisfied and, in addition, Σ0 is invariant under the group H(S) := S ∩ S−1,
then we call it a (G,S, τ)-measure space2.
(b) A Markov (G,S, τ)-measure space is a (G,S, τ)-measure space with the Markov property
E+E− = E+E0E− (cf. Definition 2.1(b)).
1Since the G-action on A is measure preserving, it defines natural representations on L1(Q,Σ, µ) and on
L2(Q,Σ, µ). However, in general we do not have an action of G on the set Q itself, but G acts naturally on the
set Σ/∼, where A ∼ B if µ(A∆B) = 0. This set corresponds to the idempotents in the algebra A. The continuity
condition (GP3) is equivalent to the continuity of the corresponding unitary representation of G on L2(Q,Σ, µ)
which in turn is equivalent to µ(gE∆E) → 0 for g → 1 and E ∈ Σ with µ(E) < ∞. See the discussion in
Appendix A or [64, p. 107] for more details.
2Note that E+θE+ ≥ 0 is equivalent to the condition that the kernel K(A,B) := µ(A∩θ(B)) on Σ+ is positive
definite.
8 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Proposition 2.9 now implies:
Proposition 3.3. For any (G,S, τ)-measure space ((Q,Σ, µ),Σ0, U, θ), we put E := L2(Q,Σ, µ),
E0 := L2(Q,Σ0, µ) and E± := L2(Q,Σ±, µ). Then the natural action of G on E defines a re-
flection positive representation of (G,S, τ). The Markov property is equivalent to the natural
map E0 → Ê being unitary and this implies that the function ϕ : S → B(E0), ϕ(s) = E0UsE0 is
multiplicative.
Definition 3.4. (a) A positive semigroup structure (H, P,A,Ω) for a symmetric semigroup
(G,S, τ) consists of
(PS1) a Hilbert space H,
(PS2) a strongly continuous ∗-representation (Ps)s∈S of (S, ]) by contractions on H,
(PS3) a commutative von Neumann algebra A on H normalized by the operators (Ph)h∈H(S),
and
(PS4) a unit vector Ω ∈ H, such that
(a) PsΩ = Ω for every s ∈ S.
(b) Ω is cyclic for the (not necessarily selfadjoint) subalgebra B ⊆ B(H) generated by A
and {Ps : s ∈ S}.
(c) For positive elements A1, . . . , An ∈ A and s1, . . . , sn−1 ∈ S, we have
〈A1Ps1A2 · · ·Psn−1AnΩ,Ω〉 ≥ 0.
(b) A standard positive semigroup structure for a symmetric semigroup (G,S, τ) consists of
a σ-finite measure space (M,S, ν) and
(SPS1) a representation (Ps)s∈S of S on L∞(M,ν) by positivity preserving operators, i.e., Psf ≥ 0
for f ≥ 0.
(SPS2) Ps1 = 1 for s ∈ S (the Markov condition).
(SPS3) P is involutive with respect to ν, i.e.,∫
M
Ps(f)hdν =
∫
M
fPs](h) dν for s ∈ S, f, h ≥ 0.
(SPS4) P is strongly continuous in measure, i.e., for each f ∈ L1(M,ν) ∩ L∞(M,ν) and every
δ > 0, s0 ∈ S, we have lim
s→s0
ν({|Psf − Ps0f | ≥ δ}) = 0.
The preceding definition generalizes the corresponding classical concepts for the case (G,S, τ)
= (R,R+,− idR) ([41] for (a) and [42] for (b)).
Remark 3.5. (a) For an operator P on L2(X,µ), the condition to be positivity preserving does
not imply that P is symmetric. In fact, if (Pf)(x) = f(Tx) for some measure preserving map
T : X → X, then P is positivity preserving and isometric. Hence we cannot expect P to be
symmetric if T is not an involution.
(b) Applying (SPS3) with h = 1, we obtain
∫
M Ps(f) dµ =
∫
M f dµ, i.e., that the measure µ
is invariant under the operators (Ps)s∈S .
(c) If P : M ×S→ [0,∞] is a Markov kernel (cf. Appendix B), then
(Pf)(x) :=
∫
M
P (x, dy)f(y)
defines a positivity preserving operator on L∞(M,S, ν) satisfying P1 = 1.
Reflection Positive Stochastic Processes Indexed by Lie Groups 9
In view of [7, Lemma 36.2], kernels on M×S are in one-to-one correspondence with additive,
positively homogeneous maps T from the convex cone of non-negative measurable functions
M → [0,∞] into itself which are Daniell continuous in the sense that lim
n→∞
T (fn) = T
(
lim
n→∞
fn
)
for monotone sequences fn ≤ fn+1.3
Remark 3.6. We consider a standard positive semigroup structure for (G,S, τ).
(a) Conditions (SPS1/2) imply that ‖Ps‖ ≤ 1 on L∞(M,ν). Further (SPS2/3) imply that the
restriction of Ps to L1 ∩ L∞ is measure preserving, and from [42, Proposition 1.2(i)] it follows
that Ps]s defines a contraction on L2(M,ν). This implies that we obtain a ∗-representation
of (S, ]) on L2(M,ν). The proof of [42, Proposition 1.2(ii)] further shows that (SPS4) implies
that this representation is strongly continuous because we have for f ∈ L1 ∩ L2:
‖Psf − Ps0f‖22 ≤
∫
{|Psf−Ps0f |≥ε}
|Psf − Ps0f |2 dν +
∫
{|Psf−Ps0f |≤ε}
|Psf − Ps0f |2 dν
≤ (2‖f‖∞)2ν({|Psf − Ps0f | ≥ ε}) + 2ε‖f‖1.
For h ∈ H(S), both operators Ph and Ph−1 = P−1
h are positivity preserving and fix 1, and
since L∞(M,ν) is a commutative von Neumann algebra, they are algebra automorphisms by
[12, Theorem 3.2.3]. The relation Ph(fg) = Ph(f)Ph(g) then leads to
PhMfP
−1
h = MPh(f)
for the multiplication operators Mf . In particular, the action of H(S) on L2(M,ν) normalizes
L∞(M,ν).
(b) If ν is a probability measure, Ω := 1 and A := L∞(M,ν), then the preceding discussion
shows that we also have a positive semigroup structure in the sense of Definition 3.4(a) for
which 1 is a cyclic vector for A. Note that Definition 3.4(b) is not a special case of (a) because
it does not require the measure ν to be finite.
The following proposition shows that the requirement that Ω is cyclic for A describes those
positive semigroup structures which are standard.
Proposition 3.7 ([41, Proposition 3.5]). Let (M,S, ν) be a probability space and (Ps)s∈S be
a positivity preserving continuous ∗-representation of S by contractions on L2(M,ν), i.e.,
Ps1 = 1 and Psf ≥ 0 for f ≥ 0, s ∈ S.
Then (L2(M), Q, L∞(M), 1) is a positive semigroup structure for which 1 is a cyclic vector
for L∞(M).
Conversely, let (H, P,A,Ω) be a positive semigroup structure for which Ω is a cyclic vector
for A. Then there exists a probability space M and a positivity preserving semigroup Q̃ on L2(M)
such that (H, P,A,Ω) ∼= (L2(M), Q̃, L∞(M), 1) as positive semigroup structures.
Proof. The first part is an immediate consequence of the definitions (see also Remark 3.6(a)),
so we only have to prove the second statement. So let (H, P,A,Ω) be a positive semigroup
structure for which Ω is a cyclic vector for A. Let M be a compact space with A ∼= C(M).
Then we obtain on M a probability measure ν, defined by∫
M
f(x) dν(x) = 〈fΩ,Ω〉 for f ∈ A.
3The notion of a positivity preserving operator on L∞(M,S, ν) is slightly weaker than this concept. In
particular, it only operates on equivalence classes of functions in L∞(X,S, ν) and not on functions itself. In
concrete situations, the positivity preserving operators actually come from kernels, which makes them easier to
deal with.
10 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Since Ω is A-cyclic, the map C(M) → H, f 7→ fΩ extends to a unitary map L2(M,ν) → H
intertwining A with the von Neumann algebra L∞(M,ν). It is clear that (SPS1-3) are satisfied
and Lemma A.1 implies the continuity in measure. �
Example 3.8 (the real oscillator semigroup). We consider the Hilbert space H = L2(Rd), with
respect to Lebesgue measure.
(a) On H we have a unitary representation by the group GLd(R) by
(Thf)(x) := |det(h)|−d/2f
(
h−1x
)
for h ∈ GLd(R), x ∈ Rd,
and we also have two representations of the abelian semigroup Symd(R)+ (the convex cone of
positive semidefinite matrices):
(a) Each A ∈ Symd(R)+ defines a multiplication operator (MAf)(x) := e−〈Ax,x〉f(x) which is
positivity preserving on L∞(Rn) but does not preserve 1; it preserves the Dirac measure δ0
in the origin.
(b) Each A ∈ Symd(R)+ specifies a uniquely determined (possibly degenerate) gaussian mea-
sure µA on Rd whose Fourier transform is given by µ̂A(x) = e−〈Ax,x〉/2. Then the convolu-
tion operator CAf := f ∗ µA is positivity preserving and leaves Lebesgue measure on Rd
invariant. For A = 1, we thus obtain the heat semigroup as (µt1)t≥0.
Any composition of these 3 types of operators Th,MA and CA is positivity preserving on
L∞(Rd), and they generate a ∗-representation of the Olshanski semigroup S := H exp(C) in
the symmetric Lie group G := Sp2d(R), where H = GLd(R), C = Symd(R)+ × Symd(R)+ ⊆
Symd(R)2 = q, and
τ
(
A B
C −A>
)
=
(
A −B
−C −A>
)
for
(
A B
C −A>
)
∈ sp2d(R)
with B> = B, C> = C (cf. Examples 2.5). The real Olshanski semigroup S is the fixed point
set an antiholomorphic involutive automorphism of the so-called oscillator semigroup SC =
Gc exp(W ) which is a complex Olshanskii semigroup [24, 30]. The elements in the interior
of S act on L2(Rn) by kernel operators with positive gaussian kernels and the elements of SC
correspond to complex-valued gaussian kernels. The semigroup S contains many interesting
symmetric one-parameter semigroups such as the Mehler semigroup e−tHosc generated by the
oscillator Hamiltonian
Hosc = −
n∑
j=1
∂2
j +
1
4
n∑
j=1
x2
j −
d
2
1, (3.1)
which fixes the Gaussian e−‖x‖
2/4.
(b) The parabolic subgroup P := Symd(R) o GLd(R)+ ⊆ Sp2d(R) is also a symmetric Lie
subgroup with h = gld(R) and q ∼= Symd(R). Here the commutative von Neumann algebra
A = L∞(Rd) is invariant under conjugation with the operators Th, so that (A, h) 7→ CATh
defines a ∗-representation of the semigroup S := Symd(R)+ o H that leads to a standard
positive semigroup structure on L2(Rd, dx).
The preceding example naturally extends to infinite-dimensional spaces as follows.
Example 3.9. Let H be a real Hilbert space.
(a) In Lemma 5.5 below we show that every continuous ∗-representation of a topological
involutive semigroup (S, ]) by contractions on a real Hilbert space H defines a standard positive
semigroup structure on the gaussian probability space (Ha, γH). Here Ha is the algebraic dual
Reflection Positive Stochastic Processes Indexed by Lie Groups 11
space of H and γH is the unique probability measure with Fourier transform γ̂H(v) = e−‖v‖
2/2
(Example 5.3).
(b) Some of the structure from Example 3.8 extends to infinite-dimensional Hilbert spaces H.
For the multiplication operator MA to be non-zero, one has to require that A ≥ 0 is trace class
(cf. [68, p. 153]). Likewise, the measures µA exist if A is trace class. In an orthonormal
basis (en)n∈N in which A is diagonal with Aen = anen, the measure µA is ⊗∞n=1γant, where
dγt(x) = (2πt)−1/2e−x
2/2t dx are the centered gaussian measures on R. For a linear operator
h ∈ GL(H) to act on L2(Ha, γH), it is necessary and sufficient that g>g−1 is Hilbert–Schmidt,
i.e., that the polar decomposition has the form g = ueX with ‖X‖2 <∞ (see also Theorem 5.7
and [52] for more details).
(c) For H = Rd, we have dγ(x) := dγH(x) = (2π)−d/2e−
‖x‖2
2 dx (Example 5.4), we obtain in
particular a standard positive semigroup structure for the contraction semigroup
C
(
Rd
)
:= {g ∈ GLd(R) : ‖g‖ ≤ 1} = Od(R) exp(−Symd(R)+),
where Symd(R)+ denotes the closed convex cone of positive semidefinite symmetric (d × d)-
matrices. The corresponding Markov operators are given by (Γ(h)f)(x) = f(h−1x) for h ∈
Od(R). Since every element s ∈ C(Rd) has a polar decomposition s = h exp(−X), X> = X ≥ 0,
diagonalization of X reduces the description of the corresponding operator to the case d = 1.
For 0 < c < 1 we have
(Γ(c)f)(x) =
∫
Rd
K̃c(x, y)f(y) dγ(y) with K̃c(x, y) =
(
1− c2
)−d/2
e
‖y‖2
2 e
− ‖cx−y‖
2
2(1−c2)
(cf. [55, p. 218]). For c = e−t, t ≥ 0, we thus obtain the Mehler semigroup, also called the
Ornstein–Uhlenbeck semigroup. It can also be described by the Mehler formula [11, Section 1.4]
(Γ(c)f)(x) =
∫
Rd
f(cx+
√
1− c2y) dγ(y). (3.2)
These operators form a hermitian strongly continuous contraction semigroup for which γ is the
unique invariant probability measure. In view of [11, Section 2.9], (3.2) still holds for infinite-
dimensional spaces.
(d) For H = Rd, it is instructive to connect the gaussian picture to the Lebesgue picture. To
this end, we note that
Φ: L2
(
Rd
)
→ L2
(
Rd, γ
)
, Φ(f)(x) = (2π)d/4e‖x‖
2/4f(x)
is a unitary isomorphism. Conjugating with Φ, we therefore obtain a ∗-representation π of the
real Olshanski semigroup S (Example 3.8) on L2(Rd, γ). This transformation has no effect on the
multiplication operators MA because they commute with Φ but it transforms the convolution
operators CA into more complicated Markov operators. For instance the Laplace operator
∆ =
∑
j ∂
2
j transforms into
∆′ := Φ∆Φ−1 = ∆ +
‖x‖2
4
−
(
E +
d
2
1
)
, where E =
∑
j
xj∂j
is the Euler operator. The unitary representation of H = GLd(R) transforms into
T ′hf = e
1
4
(‖x||2−‖h−1x‖2)(Thf)(x) = |deth|−d/2e
1
4
(‖x||2−‖h−1x‖2)f
(
h−1x
)
.
As the operators Φ and Φ−1 are positivity preserving, the semigroup S also acts by positivity
preserving operators on L2(Rn, γ). If s ∈ S preserves 1, then the transformed operator s′
preserves the Gaussian e‖x‖
2/4.
12 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
(e) The oscillator Hamiltonian Hosc from (3.1) transforms into
−∆′ := ΦHoscΦ
−1 = Φ ◦
(
−∆ +
‖x‖2
4
− d
2
1
)
◦ Φ−1 = E −∆ =
∑
j
xj∂j − ∂2
j ,
which also makes sense in the infinite-dimensional case, where ∆′ is called Umemura’s Laplace–
Beltrami operator [23, p. 221]. It is the infinitesimal generator of the Mehler semigroup et∆
′
.
Example 3.10 (a finite-dimensional example). (a) We consider the finite set M = {1, . . . , n},
endowed with the counting measure, so that L2(M,ν) ∼= Rd. Let S ⊆ GLn(R) be the subsemi-
group of invertible doubly stochastic matrices S = (Sij), i.e.,
∑
i Sij =
∑
j Sij = 1 for all i, j.
According to the Birkhoff–von Neumann theorem, S ⊆ conv(H), where H = S ∩ S−1 ∼= Sn
is the group of permutation matrices. Then S ⊆ GLn(R) a transposition stable subsemigroup
for which the action on Rn ∼= L∞(M) by matrix multiplication defines a standard positive
semigroup structure.
(b) A natural generalization of the preceding example is obtained by starting with a prob-
ability space (Q,Σ, ν) and the group H := Aut(Q,Σ, ν) of measure preserving automorphisms
of (Q,Σ). Then the probability measures µ on Q defining by Pµ(f)(x) :=
∫
H f(h−1x) dµ(h)
invertible operators on L∞(Q,Σ, ν) naturally generalizes the semigroup S ⊆ GLd(R) from (a).
3.2 Stochastic processes indexed by Lie groups
We now introduce the concept of stochastic processes where the more common index set R,
thought of as a time, is replaced by a Lie group. The forward direction is then given by a semi-
group contained in G.
Definition 3.11. Let (Q,Σ, µ) be a probability space. A stochastic process indexed by a group G
is a family (Xg)g∈G of measurable functions Xg : Q → (B,B), where (B,B) is a measurable
space, called the state space of the process.
For any such process, we obtain a measurable map Φ: Q → BG, Φ(q) = (Xg(q))g∈G with
respect to the product σ-algebra BG. Then ν := Φ∗µ is a measure on BG, called the distribution
of the process (Xg)g∈G. This measure is uniquely determined by the measures νg on Gn, obtained
for any tuple g := (g1, . . . , gn) ∈ Gn as the image of µ under the map
Xg = (Xg1 , . . . , Xgn) : Q→ Bn
(cf. [23, Section 1.3]). If g = (g) for some g ∈ G then we write νg for νg.
The process (Xg)g∈G is called stationary if the corresponding distribution on BG is invariant
under the translations
(Ugν)h := νg−1h for g, h ∈ G.
If τ ∈ Aut(G) is an automorphism, then we call the process τ -invariant if the distribution is
invariant under
(τν)h := ντ−1(h) for h ∈ G.
The following definition generalizes some concepts from G = R to general topological groups
(cf. [40, Section 2]). We only formulate the definition for real valued functions, but will use
without further comments the complex analogue where in the L2-inner products the second
factor has to be conjugated.
Reflection Positive Stochastic Processes Indexed by Lie Groups 13
Definition 3.12. (a) A linear stochastic process (indexed by the group G) is a stochastic process
(ϕ(v, g))v∈V,g∈G, where V is a real linear space, ϕ is linear in v and each ϕ(v, g) is a real valued
measurable function on a fixed probability space (Q,Σ, µ).
(b) A linear stochastic process is said to be of second order if each ϕ(v, g) is square integrable
and the stochastic process is full in the sense that, up to sets of measure zero, Σ is the smallest
σ-algebra for which all functions ϕ(v, g) are measurable.
(c) That ϕ is continuous in the quadratic mean means that V is a topological vector space,
ϕ is of second order and ϕ : V ×G→ L2(Q,Σ, µ) is continuous.
(d) Assume that ϕ is of second order. Then ϕ is (wide sense) stationary means that the
kernel
K((v, g), (v′, g′)) := E
(
ϕ(v, g)ϕ(v′, g′)
)
=
∫
Q
ϕ(v, g)ϕ(v′, g′) dµ
on V ×G satisfies the invariance condition
K((v, g), (v′, g′)) = K((v, g0g), (v′, g0g
′)) for g0, g, g
′ ∈ G, v, v′ ∈ V.
Let ϕ be a stochastic process. To adapt better to the discussion in the previous section, we
assume that ϕ(v, g) is complex-valued and that V is a complex vector space. The real case can
be treated in the same way. If ϕ is a linear stochastic process of second order, then the square
integrability of the functions ϕ(v, g) implies that the covariance kernel
K((v, g), (v′, g′)) := E
(
ϕ(v, g)ϕ(v′, g′)
)
=
∫
Q
ϕ(v, g)ϕ(v′, g′) dµ
is defined. It is a positive definite kernel on V ×G, and the map
Φ: L2(Q,Σ, µ)→ HK ⊆ CV×G, Φ(F )(v, g) := 〈F,ϕ(v, g)〉
is a partial isometry of L2(Q,Σ, µ) onto the reproducing kernel Hilbert space HK ⊆ CV×G. Its
adjoint Φ∗ is an isometric embedding of HK whose range is the closed subspace on L2(Q,Σ, µ)
generated by the (ϕ(v, g))v∈V,g∈G. Furthermore, ϕ is continuous in the quadratic mean if and
only if K is continuous.
We will from now on assume that ϕ is continuous in the quadratic mean. Then
(v, v′) := E
(
ϕ(v,1)ϕ(v′,1)
)
=
∫
Q
ϕ(v,1)ϕ(v′,1) dµ
defines a continuous positive semidefinite form on V , and in the following we may therefore
replace V by the complex Hilbert space F obtained by completing the quotient
V/{v ∈ V : (v, v) = 0}
with respect to the induced norm. We thus obtain a stochastic process ϕ(w, g) on F ×G. Now
we may consider F as a closed subspace of L2(Q,Σ, µ) and write E0 : L2(Q,Σ, µ) → F for the
orthogonal projection.
If K is continuous and F is a complex Hilbert space, then K also defines a positive definite
B(F)-valued kernel
KF : G×G→ B(F),
〈
KF (g, h)v, w
〉
= K((v, h), (w, g)),
so that we can realize the closed subspace of L2(Q,µ) generated by the ϕ(v, g) as the reproducing
kernel Hilbert space HKF ⊆ CF of F-valued functions on G. Here F is isomorphic to the closed
subspace of L2(Q,Σ, µ) generated by the functions (ϕ(v,1))v∈V .
14 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Consider now (d) and assume that ϕ is continuous in the quadratic mean. This condition
ensures that we obtain a unitary representation (Ug)g∈G on the subspace HK ⊆ L2(Q,µ) satis-
fying
(Ugϕ)(v, h) = ϕ(v, gh), v ∈ V, g, h ∈ G.
Under assumption (c), we thus obtain a strongly continuous B(F)-valued positive definite func-
tion r(g) := E0UgE0. The invariance condition leads to a Sesq(V )-valued positive definite
function on G as discussed in [51], but we will not follow up on that in this article.
Example 3.13. Let (G, τ) be a symmetric Lie group and (Xg)g∈G be a stationary, τ -invariant,
full stochastic process on (Q,ΣQ, µQ). Then its distribution (BG,BG, ν) satisfies the conditions
(GP1,2,4,5) of a (G, τ)-probability space with respect to the natural actions of G and τ on BG,
where Σ0 is the σ-algebra generated by (Xh)h∈H , i.e., the smallest subalgebra for which these
functions are measurable. In this context (GP3) is equivalent to the continuity of the unitary
representation of G on L2(BG,BG, ν). We refer to Appendix A for a more detailed discussion
of the continuity condition.
Example 3.14 (cf. Proposition 3.3). Suppose that ((Q,Σ, µ),Σ0, U, θ) is a (G, τ)-probability
space, so that we obtain on E := L2(Q,Σ, µ) a unitary representation of G. The subspace
E0 := L2(Q,Σ0, µ) is cyclic under the algebra B generated by (Ug)g∈G and the multiplication
operators Mf , f ∈ L∞(Q,Σ0, µ). The prescription
ϕ(v, g) := π(g)v, g ∈ G, v ∈ E0
defines a wide sense stationary process indexed by E0 ×G.
The scalar product on L2(Q,Σ, µ) is completely determined by the n-point functionals:
〈Ug1Mf1 · · ·Mfn−1UgnMfnv, w〉E0 , v, w ∈ E0, gj ∈ G, fj ∈ L∞(Q,Σ0, µ),
resp.,
〈Ug1Mf1 · · ·Ugnfn, f0〉E0 = E
(
f0ϕ(f1, g1)ϕ(f2, g1g2) · · ·ϕ(fn, g1 · · · gn)|Σ0
)
,
or the operators
E0Ug1Mf1 · · ·Mfn−1UgnE0, gj ∈ G, fj ∈ L∞(Q,Σ0, µ).
In general E0 is not cyclic for G, so the process ϕ(v, g) provides a means to construct the
whole space L2(Q,Σ, µ) in the spirit of a GNS construction for B.
Example 3.15. In [64] free euclidean fields (scalar of positive mass m > 0) are dealt with as
gaussian processes indexed by the Hilbert space Nm defined by the norm
‖f‖2Nm = 2
∫
Rd+1
|f̂(k)|2
k2 +m2
d2k for f ∈ S
(
Rd+1
)
,
so that the corresponding euclidean Hilbert space is the Fock space Γ(Nm) (see Section 5.1).
Here Nm carries a unitary representation of the euclidean group E(d) = Mot(Rd), and since the
time translation group has continuous spectrum, the corresponding action on Γ(Nm) is ergodic
by Theorem 5.26 below (for time translations, space translations and the full euclidean group).
[64, Theorem III.6] provides a Feynman–Kac–Nelson formula for the gaussian free field. In
this context, Nelson’s main achievement in [54] was that he obtained a manifestly euclidean
invariant path integral.
Reflection Positive Stochastic Processes Indexed by Lie Groups 15
3.3 Associated positive semigroup structures and reconstruction
Our goal in this section is to prove the reconstruction theorem (Theorem 3.18). This is a central
result which asserts that positive semigroup structures all come from (G,S, τ)-measure spaces.
We start with the following lemma which is an adaption of [41, Lemma 2.2] and [42, Corol-
lary 1.5] to (G,S, τ).
Lemma 3.16. Let ((Q,Σ, µ),Σ0, U, θ) be a (G,S, τ)-measure space, A := L∞(Q,Σ0, µ), E :=
L2(Q,Σ, µ), E0 := L2(Q,Σ0, µ), and q : E+ := L2(Q,Σ+, µ) → Ê be the canonical map. Then
the following assertions hold:
(a) For f ∈ A, let Mf denote the corresponding multiplication operator on E. Then there
exists a bounded operator M̂f ∈ B(Ê) with q ◦Mf |E+ = M̂f ◦ q and ‖M̂f‖ = ‖f‖∞.
(b) π(f) := M̂f is a faithful weakly continuous representation of the commutative von Neu-
mann algebra A on Ê.
(c) In the Markov case we identify Ê with E0 and q with E0 (Proposition 3.3). For g1 ≺S
g2 ≺S · · · ≺S gn in G, non-negative functions f1, . . . , fn ∈ A and fgj := Ugjfj, we have∫
Q
fg1 · · · fgn dµ =
∫
Q
M̂f1Ûg−1
1 g2
· · · M̂fn−1Ûg−1
n−1gn
M̂fn1 dµ.
If, in addition, µ is finite and positive, then Ω := µ(Q)−1/2q(1) satisfies:
(d) For g1 ≺S g2 ≺S · · · ≺S gn in G, f1, . . . , fn ∈ A and fgj := Ugjfj, we have∫
Q
fg1 · · · fgn dµ =
〈
M̂f1Ûg−1
1 g2
· · · M̂fn−1Ûg−1
n−1gn
M̂fnΩ,Ω
〉
.
(e) Ω is a separating vector for A and ÛsΩ = Ω for every s ∈ S.
(f) Ω is cyclic for the algebra B generated by A and (Ûs)s∈S.
Proof. (a, b) Since θ commutes with E0, it preserves the subspace E0 := L2(Q,Σ0, µ). Reflec-
tion positivity and E0 ⊆ E+ now imply that θ(f) = f for f ∈ E0. The preceding argument implies
that Mf commutes with θ. As Mf and its adjoint M∗f = Mf preserve E±, these operators also
preserve N := E+ ∩ θ(E−)⊥, so that we obtain a well-defined operator M̂f : q(E+) → Ê . For
v ∈ E+, we then have∥∥M̂f v̂
∥∥2
= 〈θMfv,Mfv〉 = 〈θM|f |2v, v〉 ≤ ‖f‖2∞‖v‖2.
Applying this estimate to the functions (fn)n∈N0 , we get ‖M̂f‖ ≤ ‖f‖∞ from [48, Lemma II.3.8].
Since q is isometric on E0 and ‖Mf |E0‖ = ‖f‖∞, we actually obtain the equality ‖M̂f‖ = ‖f‖∞.
This proves (a) and (b).
(c) ([42, Corollary 1.5]) First we observe that the operator
Mfg1
· · ·MfgnUgn = MUg1f1
· · ·MUgnfnUgn = Ug1Mf1Ug−1
1 g2
Mf2 · · ·Mfn−1Ug−1
n−1gn
Mfn
on E+ is intertwined by q with Ûg1M̂f1Ûg−1
1 g2
· · · Ûg−1
n−1gn
M̂fn . Applying this operator to the
constant function 1 and integrating yields the assertion.
From now on we assume that 0 < µ(Q) <∞.
16 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
(d) With the same argument as under (c), we derive∫
Q
fg1 · · · fgn dµ = 〈Mfg1
· · ·Mfgn1, 1〉
= 〈θMfg1
· · ·MfgnUgn1, 1〉 =
〈
M̂f1Ûg−1
1 g2
· · · M̂fn−1Ûg−1
n−1gn
M̂fnΩ,Ω
〉
.
(e) The relation ÛsΩ = Ω is a trivial consequence of Us1 = 1 for s ∈ S. Since q|E0 is
isometric and intertwines the representation of A on E0 with the representation of A on Ê , the
A-separating vector 1 ∈ E0 is mapped by q to an A-separating vector.
(f) Since E0 is US-cyclic in E+, this follows from the fact that 1 is A-cyclic in L2(Q,Σ0, µ). �
Definition 3.17. The preceding lemma shows that, if ((Q,Σ, µ),Σ0, U, θ) is a finite (G,S, τ)-
measure space, then (Ê , Û ,A, q(1)) is a positive semigroup structure for
A =
{
M̂f : f ∈ L∞(Q,Σ0, µ)
}
.
We call it the associated positive semigroup structure.
We now come to our version of Klein’s reconstruction theorem. Note that every discrete group
is in particular a 0-dimensional Lie group, so that the following theorem applies in particular to
discrete groups.
Theorem 3.18 (Reconstruction Theorem). Let (G, τ) be a symmetric Lie group and S ⊆ G be
a ]-invariant subsemigroup satisfying G = S ∪S−1. Then every positive semigroup structure for
(G,S, τ) is associated to some (G,S, τ)-probability space ((Q,Σ, µ),Σ0, U, θ).
Proof. We follow the proof for (G,S, τ) = (R,R+,− id) in [41, Theorem 2.4]. In view of
the Gelfand representation theorem, there exists a compact space Q0 with A ∼= C(Q0) as C∗-
algebras. Then Q := QG0 , the space of all functions ω : G→ Q0, is compact with respect to the
product topology. The group G acts on Q by
(Ugω)(x) := ω
(
g−1x
)
and, accordingly, on functions on Q by (UgF )(ω) = F (Ug−1ω). We further obtain an involution
(θω)(g) := ω(τ(g)) satisfying
θUgθ = Uτ(g) for g ∈ G.
Let Σ0 be the smallest σ-algebra on Q for which X1(ω) := ω(1) is measurable, Σg := UgΣ1,
and let Σ be the σ-algebra generated by the Σg, g ∈ G. To construct a measure µ on Σ, it
suffices to construct a Radon measure on Q, i.e., a positive functional on the C∗-algebra C(Q).
By continuity, it actually suffices to do this on a dense unital subalgebra. Such a subalgebra is
generated by the G-translates of functions in Ξ := C(Q0, {0, 1}), which is a generating subset
of A. Now let R ⊆ C(Q) be the subspace spanned by functions of the form
F (ω) := χ1(ω(g1)) · · ·χn(ω(gn)), n ∈ N0, χj ∈ Ξ, gj ∈ G, i.e., F =
n∏
j=1
Ugjχj ,
when we identify C(Q0) with the subalgebra X∗1C(Q0) of C(Q). Clearly, R is a subalgebra of
C(Q), and the Stone–Weierstraß theorem implies that it is dense.
Since G = S ∪ S−1, every finite subset of G is of the form {g1, . . . , gn} with g1 ≺S · · · ≺S gn,
i.e., sj := g−1
j gj+1 ∈ S for j = 1, . . . , n− 1. Next we show that that, for (Aj , gj) ∈ A ×G with
g1 ≺S · · · ≺S gn, the operator
A1Ps1A2 · · ·Psn−1An (3.3)
Reflection Positive Stochastic Processes Indexed by Lie Groups 17
does not depend on the way we enumerate the gj . Suppose that we also have gj+1 ≺S gj for
some j. Then h := sj = g−1
j gj+1 ∈ H(S) = S ∩ S−1. We put g̃i := gi for i 6= j, j + 1 and set
g̃j := gj+1, g̃j+1 = gj . Accordingly, we put Ãj := Aj+1 and Ãj+1 := Aj . This leads to
s̃i =
si for i 6= j − 1, j, j + 1,
sj−1sj = sj−1h for i = j − 1,
h−1 for i = j,
sjsj+1 = hsj+1 for i = j + 1.
We thus obtain
Ps̃j−1
ÃjPs̃j Ãj+1Ps̃j+1
= Psj−1hAj+1P
−1
h AjPhsj+1
= Psj−1
(
PhAj+1P
−1
h
)
AjPhPsj+1
= Psj−1Aj
(
PhAj+1P
−1
h
)
PhPsj+1 = Psj−1AjPsjAj+1Psj+1 ,
where we use that PhAj+1P
−1
h ∈ A, H(S) normalizes A, and the commutativity of A. The
above argument shows the asserted independence of (3.3) because any increasing rearrangement
can be obtained by successive transpositions.
For n ∈ N and s1, . . . , sn−1 ∈ S, the map
An ∼= C(Q0)n → R, (A1, . . . , An) 7→ 〈A1Ps1A2 · · ·Psn−1AnΩ,Ω〉
is n-linear, so that there exists a well-defined linear functional ρ : R → R satisfying
ρ
n∏
j=1
Ugjfj
=
〈
f1Pg−1
1 g2
f2 · · ·Pg−1
n−1gn
fnΩ,Ω
〉
(3.4)
for f1, . . . , fn ∈ span Ξ, g1 ≺S · · · ≺S gn, n ∈ N0.
To see that ρ is positive, we observe that every non-negative function F ∈ R can be written
in the form
F =
∑
k
ckFk with Fk :=
n∏
j=1
Ugjχj,k, where FkFk′ = 0 for k 6= k′.
Then ck ≥ 0 for all k, so that
ρ(F ) =
∑
k
ckρ
n∏
j=1
Ugj,kχj,k
≥ 0
by (3.4). This proves that ρ ≥ 0, and thus, by continuity, it extends to a positive functional
on C(Q), which defines a Radon measure µ on Q.
Next we show that ((Q,Σ, µ),Σ0, P, θ) is a (G,S, τ)-measure space. The G-invariance of the
right hand side of (3.4) implies that G acts on Q by measure preserving transformations. To see
that µ is also θ-invariant, we first note that g ≺S h implies τ(h) ≺S τ(g) because τ(S) = S−1.
We thus obtain
θ
n∏
j=1
Ugjfj
=
n−1∏
j=0
Uτ(gn−j)fn−j .
Hence the θ-invariance of µ follows from〈
f1Pg−1
1 g2
f2 · · ·Pg−1
n−1gn
fnΩ,Ω
〉
=
〈
Ω, fnPτ(gn)−1τ(gn−1) · · · f2Pτ(g2)−1τ(g1)f1Ω
〉
=
〈
fnPτ(gn)−1τ(gn−1) · · · f2Pτ(g2)−1τ(g1)f1Ω,Ω
〉
.
18 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
It remains to verify 〈θF, F 〉 ≥ 0 for F ∈ L2(Q,Σ+, µ). The functions of the form
fg1 · · · fgn = Ug1Mf1Ug−1
1 g2
· · ·Ug−1
n−1gn
MfnU
−1
gn 1, fj ∈ A,1 ≺S g1 ≺S g2 ≺S · · · ≺S gn,
form a total subset of E+. On pairs of such functions we have〈
θfg1 · · · fgn , f ′h1 · · · f
′
hm
〉
=
〈
θUg1Mf1Ug−1
1 g2
· · ·Ug−1
n−1gn
fn, Uh1Mf ′1
Uh−1
1 h2
· · ·Uh−1
m−1hm
f ′m
〉
=
〈
Uτ(g1)Mf1Uτ(g1)−1τ(g2) · · ·Uτ(gn−1)−1τ(gn)fn, Uh1Mf ′1
Uh−1
1 h2
· · ·Uh−1
m−1hm
f ′m
〉
=
〈
1,MfnUτ(gn)−1τ(gn−1) · · ·Uτ(g2)−1τ(g1)Mf1Uτ(g1)−1Uh1Mf ′1
Uh−1
1 h2
· · ·Uh−1
m−1hm
f ′m
〉
=
〈
Ω, M̂fnPτ(gn)−1τ(gn−1) · · ·Pτ(g2)−1τ(g1)M̂f1Pτ(g1)−1Ph1M̂f ′1
Ph−1
1 h2
· · ·Ph−1
m−1hm
M̂f ′mΩ
〉
=
〈
Pg1M̂f1Pg−1
1 g2
· · ·Pg−1
n−1gn
M̂fnΩ, Ph1M̂f ′1
Ph−1
1 h2
· · ·Ph−1
m−1hm
M̂f ′mΩ
〉
.
This implies that〈
θ
∑
j
cjfg1,j · · · fgn,j ,
∑
k
ckfg1,k · · · fgn,k
〉
=
∥∥∥∑
j
cjPg1,jM̂f1,jPg−1
1,j g2,j
· · ·Pg−1
n−1,jgn,j
M̂fn,jΩ
∥∥∥2
≥ 0.
We conclude that E+ is θ-positive and that the canonical map q : E+ → Ê is equivalent to the
map
q : E+ → H, fg1 · · · fgn 7→ Pg1M̂f1Pg−1
1 g2
· · ·Pg−1
n−1gn
M̂fnΩ.
This map satisfies q ◦Mf = M̂f ◦ q and q ◦ Us = Ps ◦ q for s ∈ S. Therefore (H, P,A,Ω) is
equivalent to the associated positive semigroup structure. �
Remark 3.19. Theorem 3.18 implies in particular that, for every positive semigroup structure,
Ω is separating for A [41, Remark 2.5].
Remark 3.20. We have seen in the proof of Theorem 3.18 that it is sometimes more convenient
to deal with the unital W ∗-algebra A = L∞(Q,Σ0, ν) as a C∗-algebra and write it as C(Q0) for
a compact (Hausdorff) space Q0. Then the Riesz representation theorem asserts that positive
functionals on C(Q0) correspond to regular Borel measures µ on Q0. On the other hand, the
restriction of any such measure to the σ-algebra of Baire sets, i.e., the smallest σ-algebra for
which all continuous functions are measurable, already determines the corresponding linear
functional on C(Q0) by integration. Therefore the positive functionals on C(Q0) are in one-to-
one correspondence with finite positive Baire measures on Q0.
Since the set Σ0/Jν of equivalence classes modulo ν-null sets corresponds to the idempotents
in A, resp., C(Q0), Baire measures on Q0 correspond to measures on (Q,Σ0) that are absolutely
continuous with respect to ν.
A slight variation of the argument in the proof of Theorem 3.18 leads to the following generali-
zation of the Klein–Landau reconstruction theorem [42, Theorem 1.8] which deals with standard
positive semigroup structures. For the proof of this theorem, we need a suitable version of [42,
Lemma 1.9] to deal with the case where the measure ν is infinite.
Reflection Positive Stochastic Processes Indexed by Lie Groups 19
Lemma 3.21. Let (G,S, τ) be a symmetric semigroup with G = S ∪ S−1. Let Q0 be a compact
Hausdorff space and ν0 a σ-finite measure on the Baire sets in Q0. Suppose that (Ps)s∈S is
a positivity preserving semigroup acting on C(Q0) leaving the function 1 invariant. Then there
exists a unique σ-finite Baire measure µ on the compact Hausdorff space Q = QG0 , such that, for
τ(sm) ≺S · · · ≺S τ(s1) ≺S 1 ≺S t1 ≺S · · · ≺S tn
and non-negative g1, . . . , gm, f1, . . . , fn ∈ C(Q0), we have∫
Q
gm(ϕ(τ(sm))) · · · g1(ϕ(τ(s1))f1(ϕ(t1)) · · · fn(ϕ(tn)) dµ(ϕ)
=
∫
Q0
(
Ps1Mg1Ps−1
1 s2
Mg2 · · ·Ps−1
m−1sm
Mgm1
)(
Pt1Mf1Pt−1
1 t2
Mf2 · · ·Pt−1
n−1tn
Mfn1
)
dν(q0),
where, as before, Mf denotes multiplication by f .
Proof. First we assume that the measure ν is finite. Let R ⊆ C(Q) be as in the proof of
Theorem 3.18. We then obtain a positive functional ρ on R, which is uniquely determined by
the requirement that, for
F (q) = gm(ϕ(τ(sm))) · · · g1(ϕ(τ(s1)))f1(ϕ(t1)) · · · fn(ϕ(tn)),
we have
ρ(F ) =
∫
Q0
(
Ps1Mg1Ps−1
1 s2
Mg2 · · ·Ps−1
m−1sm
Mgm1
)(
Pt1Mf1Pt−1
1 t2
Mf2 · · ·Pt−1
n−1tn
Mfn1
)
dν(q0)
(see the proof of [42, Lemma 1.9] for details). The Riesz–Markov theorem now leads to a unique
finite Baire measure µ on Q with
∫
Q F dµ = ρ(F ) for F ∈ R.
If ν is not finite, we write it as ν =
∑
j νj , where the measures νj are finite. If µj denotes the
measure on Q corresponding to νj , the measure µ :=
∑
j µj satisfies all our requirements. �
Theorem 3.22 (Reconstruction Theorem – Standard Case). If (G,S, τ) is a symmetric semi-
group with G = S ∪ S−1, then every standard positive semigroup structure for (G,S, τ) is asso-
ciated to some (G,S, τ)-measure space ((Q,Σ, µ),Σ0, U, θ) which is unique up to G-equivariant
isomorphism of measure spaces.
Proof. First we show uniqueness. Let ((Q′,Σ′, µ′),Σ′0, U
′, θ′) be a Markov (G,S, τ)-measure
space for which (L2(M,ν), P, L∞(M,ν)) is the associated standard positive semigroup structure.
Lemma 3.16(c) implies that the integrals of products of the form fg1 · · · fgn , g1 ≺S · · · ≺S gn
are uniquely determined by the relation∫
Q
fg1 · · · fgn dµ′ =
∫
Q
Mf1Pg−1
1 g2
· · ·Mfn−1Pg−1
n−1gn
Mfn1 dν.
Since such products for fj ∈ L2(M,ν) ∩ L∞(M,ν) form a total subset of L2(Q′,Σ′, µ′), any
isomorphism L2(Q′,Σ′0, µ
′)→ L2(M,ν) ∼= L2(Q,Σ0, µ) extends to a G-equivariant isomorphism
L2(Q′,Σ′, µ′)→ L2(Q,Σ, µ).
For the existence of the measure µ, we now use Lemma 3.21. Here we write Q0 for the
spectrum of the C∗-algebra A := L∞(M,ν), so that A ∼= C(Q0) and ν defines a Baire measure
on Q0. We may then identify (M,S, ν) with (Q0,Σ0, ν), where Σ0 is the σ-algebra of Baire
subsets of Q0.
20 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Now let µ be the Baire measure on Q := QG0 from Lemma 3.21. Then the relation P ∗s = Ps] ,
as operators on L2(M,ν), leads to∫
Q
fg1 · · · fgn dµ =
∫
Q
Mf1Pg−1
1 g2
· · ·Mfn−1Pg−1
n−1gn
Mfn1 dν
for 0 ≤ f1, . . . , fn ∈ C(Q0) and g1 ≺S · · · ≺S gn. The same calculations as in the proof of
Theorem 3.18 now show that µ is invariant under G and θ. �
Remark 3.23. In the standard case, it is interesting to take a closer look at the structure of
the measure space (Q,Σ, µ). Then H ∼= L2(M,S, ν), and for L∞(M,S, ν) ∼= C(Q0) we obtain
on the compact space Q := QG0 a measure which is determined by the corresponding cylinder
measures:∫
Q
fg1 · · · fgn dµ =
∫
M
Mf1Pg−1
1 g2
· · ·Mfn−1Pg−1
n−1gn
Mfn1 dν (3.5)
for 0 ≤ f1, . . . , fn ∈ C(Q0) and g1 ≺S · · · ≺S gn.
For every g ∈ G, we then have a continuous projection qg : Q → Q0, so that we obtain
a homomorphism q∗g : L∞(M,S, ν) ∼= C(Q0)→ C(Q) of C∗-algebras. This leads to a morphism
S/Jν → Σ/Jµ of Boolean σ-algebras. At this point it is a natural question whether this
morphism can be realized by a measurable function fg : Q → M . In [66, Theorem 1.6] this is
answered in the affirmative for M = Rn, but this implies the corresponding result for completely
metrizable separable spaces, because these spaces can be realized as closed subsets of some RN.
Once we have the measurable maps fg : Q → M , they combine to a stochastic process (fg)g∈G
with state space (M,S), so that we obtain an isomorphism of measure spaces f : (Q,Σ, µ) →
(MG,SG, µ′).
If (M,S, ν) is a polish space, i.e., M carries a topology for which it is completely metrizable
and separable and S is the σ-algebra of Borel sets, then [7, Corollary 35.4] directly implies the
existence of a Borel measure µ′ on the measurable space (MG,SM ) with the correct projections
onto finite products:∫
MG
f1(ω(g1)) · · · fn(ω(gn)) dµ′(ω) =
∫
Q
Mf1Pg−1
1 g2
· · ·Mfn−1Pg−1
n−1gn
Mfn1 dν
for 0 ≤ f1, . . . , fn ∈ C(Q0) and g1 ≺S · · · ≺S gn. Therefore we obtain a realization of our
(G,S, τ)-measure space on (MG,SG, µ′).
Remark 3.24. (a) Consider the case (G,S, τ) = (R,R+,− idR) and assume that the opera-
tors Ptf are obtained from Markov kernels Pt : M ×S→ [0,∞] in the sense that
(Ptf)(x) =
∫
M
Pt(x, dy)f(y)
(cf. Appendix B). In view of Remark B.3, the measures P νt can be written as
P νt = νPt, i.e., P νt (A) =
∫
M
ν(dx)Pt(x,A)
for the kernel
Pt(x0, B) =
∫
Mn
χB(x1, . . . , xn)Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn)
on M ×Sn. In particular, we have for each x ∈M a probability measure Px on{
ω ∈MR+ : ω(0) = x
}
,
Reflection Positive Stochastic Processes Indexed by Lie Groups 21
defining a kernel P : M ×SR+ → [0,∞], such that
P ν = νP, i.e., P ν(A) =
∫
M
ν(dx)P (x,A).
For any t ≥ 0, we then have
Pt(x0, B) =
∫
MR+
χB(ω(t))P (x0, dω) = P (x0, {ω(t) ∈ B}),
which leads to
(Ptf)(x) =
∫
M
Pt(x, dy)f(y) =
∫
MR+
P (x, dω)f(ω(t)). (3.6)
This is an abstract version of the Feynman–Kac–Nelson formula that expresses the value of Ptf
in x ∈ M as an integral over all paths [0, t] → M starting in x with respect to the probability
measure Px.
We also note that, for t > 0, we have the relation∫
MR
f(ω(t)) dP ν(ω) =
∫
Q
f dν for t ∈ R,
and, for t < s,∫
M
∫
M
f1(x)ν(dx)f2(y)Ps−t(x, dy) =
∫
M
f1(x)(Ps−tf2)(x) dν(x)
=
∫
MR
f1(ω(t))f2(ω(s)) dP ν(ω).
(b) In the special case where M = G is a topological group and Ptf = f ∗ µt, we have
(Ptf)(x) =
∫
G
f(xy) dµt(y) =
∫
G
Pt(x, dy)f(y) for Pt(x,A) = µt(x
−1A).
Therefore the operators Pt are given by Markov kernels. Here, the measurability of the functions
x 7→ µt(x
−1A) =
∫
G
χA(xy) dµt(y)
follows from the measurability of the function (x, y) 7→ χA(xy).
Let P (G) be the path group GR and let P∗(G) be the subgroup of pinned paths
P∗(G) = {ω ∈ P (G) : ω(0) = 1}.
We have the relations∫
P (G)
f1(ω(0))f2(ω(t)) dµ(ω) =
∫
G
∫
G
f1(g1)f2(g1g2) dν(g1)dµt(g2) for t > 0
and ∫
P∗(G)
f(ω(t)) dP ν(ω) =
∫
G
f(g) dµt(g).
This leads for f ∈ L2(G, ν) and t ≥ 0 to
(Ptf)(x) = (f ∗ µt)(x) =
∫
G
f(xg) dµt(g) =
∫
GR+
f(xω(t)) dP (ω).
This is a group version of the Feynman–Kac–Nelson formula (3.6) which expresses (Ptf)(x) as
an integral over all paths ω : [0, t]→ G with ω(0) = x.
22 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Remark 3.25. The assumption that G = S ∪ S−1 is very restrictive (see the discussion in
Subsection 3.5). We mainly use it to ensure that the order ≺S on G is total, which is a crucial
ingredient in the construction. However, if S ⊆ G is a subsemigroup not satisfying the totality
condition, i.e., S ∪ S−1 6= G, then one may still consider subsets C ⊆ G on which the order ≺S
is total, so-called C-chains. Typical examples of such ≺S-chains are curves C = γ(I), I ⊆ R,
with γ(t2) ∈ γ(t1)S for t1 < t2 in I. For any such chain C, one can still construct a stochastic
process indexed by C, but we then don’t have the symmetries implemented by a G-action on
the corresponding probability space.
Remark 3.26. The Abel–Klein reconstruction theorem leads in particular to realizations of
dilations of the corresponding standard positive semigroup structures (Pt)t≥0 of the following
kind:
(a) Positivity preserving semigroups obtained by integrating a measure preserving action of
G on some measure space (X,S, ν) to operators
(Ptf)(x) :=
∫
G
(g.f)(x) dµt(g),
where the measures µt on G are symmetric. If (X,S) is polish, the corresponding measure
space can be realized on the path space XR. Here the requirement that G acts continuously
on L2(X,S, ν) implies that (Pt)t≥0 is strongly continuous on L2(X,S, ν) (see (b) below
for a more general argument) and therefore continuous in measure (Remark 3.6(c)).
(b) Semigroups of the type (Γ(π(µt)))t≥0, where (π,H) is a continuous unitary representation
of G on H and Γ is the functor of second quantization (see Section 5.1 for more infor-
mation about second quantization). Here we use that, for every continuous orthogonal
representation (π,H) of G and a Borel probability measure ν on G, we can define
π(ν) :=
∫
G
π(g) dν(g)
as the operator representing the hermitian form
q(v, w) :=
∫
G
〈π(g)v, w〉 dν(g)
which satisfies |q(v, w)| ≤ ‖v‖‖w‖ for v, w ∈ H. For any one-parameter semigroup (µt)t≥0
of symmetric probability measures, we thus obtain a hermitian one-parameter semigroup
of contractions (π(µt))t≥0. The continuity of this one-parameter semigroup follows from
the continuity of the functions
t 7→ 〈π(µt)v, w〉 =
∫
G
〈π(g)v, w〉 dµt(g),
which is a consequence of the weak convergence µt → δ1 for t → 0 and the boundedness
of the matrix coefficients of π.
3.4 Klein’s characterization of the Markov case
The following theorem characterizes the positive semigroup structures arising from Markov path
spaces as those for which Ω is a cyclic vector for A, which is considerably stronger than condition
(PS4)(b).
Theorem 3.27 ([41, Theorem 3.1]). Let ((Q,Σ, µ),Σ0, U, θ) be a (G,S, τ)-probability space and
let (Ê , Û ,A,Ω) be its associated positive semigroup structure. Then ((Q,Σ, µ),Σ0, U, θ) is Markov
if and only if Ω is cyclic for A.
Reflection Positive Stochastic Processes Indexed by Lie Groups 23
Proof. The Markov property is equivalent to q(E0) = Ê (Proposition 3.3). Since E0 = A · 1,
this condition is equivalent to Ω = q(1) being A-cyclic in Ê . �
Remark 3.28 (cf. [41, Remark 3.3]). Suppose that (H, P,A,Ω) is a positive semigroup structure
for which Ω is A-cyclic, so that Proposition 3.7 implies that H ∼= L2(M) for a probability
space M . Then condition (PS4)(c) follows from the seemingly weaker condition
A,B ∈ A+ := {C ∈ A : 0 ≤ C}, s ∈ S ⇒ 〈AΩ, PsBΩ〉 ≥ 0.
In fact, in this case A+Ω is dense in the closed cone L2(Q,Σ0, µ)+ of non-negative functions,
so that 〈AΩ, PsBΩ〉 ≥ 0 for all A,B ∈ A+ implies that the operator Ps on L2(Q,Σ0, µ) is
positivity preserving. Then
Ps1A1 · · ·PsnAnΩ ≥ 0
for Aj ∈ A+ and sj ∈ S, and the assertion follows.
Remark 3.29. (a) We consider the Markov case where AΩ is dense in H, so that E0
∼=
L2(Q,Σ0, µ) is a G-cyclic subspace of E . Then ϕ(g) = E0UgE0 satisfies ϕ(s) = Ûs = Ps
for s ∈ S. Therefore the G-invariant subspace of E generated by E0 carries a representation
which is a unitary dilation of the representation (P,H) of S (cf. Remark 2.8).
(b) The set of finite products fg1 · · · fgn , gj ∈ G, fj ∈ E0
∼= L∞(M,S, ν)∩L1(M,S, ν) spans
a dense subspace of E . Accordingly, the restriction gj ∈ S leads to a dense subspace of E+. This
permits us to give sufficient conditions for E+ to be cyclic in E . Since
Ugfg1 · · · fgn = fgg1 · · · fggn ,
this is the case if, for every finite subset F ⊆ G, there exists a g ∈ G with gF ⊆ S, i.e.,
g−1 ≺S F . This means that the order ≺S on G is filtered, i.e., for g1, g2 ∈ G, there exists
a g3 ∈ G with g3 ≺S g1, g2. This property is easily seen to be equivalent to G = S−1S because
g−1
1 g2 ∈ S−1S is equivalent to the existence of s1, s2 ∈ S with g3 := g2s
−1
2 = g1s
−1
1 ≺S g1, g2
(cf. [25]). We conclude that E+ is G-cyclic in E if G = S−1S. Note that this is in particular the
case if G = S ∪ S−1.
3.5 Total subsemigroups of Lie groups
In this section we briefly discuss total subsemigroups. Our standard references are [25, Chap-
ter 6] and [43]. For Lie groups G, subsemigroups S with dense interior satisfying G = S ∪ S−1
are rare. Typical examples are:
(a) S = [0,∞[= R+ ⊆ G = R.
(b) S = R+ oR×+ ⊆ G = Ro R×+ (the ax+ b-group).
(c) A subsemigroup S ⊆ G = S̃L2(R) (the simply connected covering group of SL2(R)), whose
boundary is a 2-dimensional subgroup isomorphic to the ax+ b-group. Actually G is the
simply connected covering of the projective group PSL2(R) acting by orientation preserving
projective maps on the projective line P(R2) ∼= S1. Identifying S̃1 with R, we have that
S = {g ∈ G : g(0) ≥ 0} = {g ∈ G : g(R+) ⊆ R+}.
In terms of an Iwasawa decomposition G = KAN with K ∼= R and K+ := R+, we have
S = K+AN = ANK+ and R ∼= K ∼= G/AN .
24 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
It is easy to see that S is contained in a maximal subsemigroup of S and if g/ rad(g) is
a compact Lie algebra, then the classification of maximal subsemigroups M [25, Theorem 6.18]
implies that ∂M is a codimension 1 subgroup of G. According to the classification of hyperplane
subalgebras of finite-dimensional Lie algebras [27, 28], this means that, if N ⊆ ∂M is the largest
connected normal subgroup of G contained in ∂M , then M/N ⊆ G/N is one of the three types
described above. The examples under (a)–(c) are also semigroups whose boundary ∂S is a sub-
group, so that S0 is well bounded in the sense of [16, 59]. Such a semigroup, for which ∂S
contains no non-trivial connected normal subgroup, is isomorphic to (a), (b) or (c) [16, 59].
From [50] we know that, for the first two types (a) and (b), there are non-trivial reflection
positive representations for (G,S, τ), where τ is such that Gτ = ∂S. On sl2(R), the fixed point
set of any involutive automorphism is 1-dimensional, hence cannot be a hyperplane. Suppose
that, for G = S̃L2(R), Gτ ⊆ ∂S. Then S is invariant under s 7→ s] = τ(s)−1, but one can
show that all involutive continuous representations of (S, ]) are trivial. This follows from the
Lüscher–Mack theorem [46] and the fact that all involutive contractive representations of the
group (Aff(R), ]) must be trivial on the subgroup {0} o R×+. This is a consequence of the fact
that if, for a selfadjoint operator A, all operators etA are contractions, we have Spec(A) = {0},
hence A = 0.
Example 3.30. S = Rd+ ⊆ G = Rd satisfies G = S ∪ S−1. Since this subsemigroup is
invariant under Od−1(R), we likewise obtain G1 = S1 ∪ S−1
1 for G1 := Rd o Od−1(R) and
S1 := Rd+oOd−1(R). Note that (G1, τ1) with τ1(b, A) := (θ(b), A) is a symmetric Lie group with
one-dimensional q.
3.6 The discrete case
Our reconstruction theorem applies in particular to (G,S, τ) = (Z,N0,− idZ). Here we may
start with a Markov operator P on a polish probability space (M,S, ν), so that P is positi-
vity preserving and satisfies P1 = 1 (cf. Appendix B). If P defines a symmetric operator on
L2(M,S, ν), then we obtain a unitary operator U acting by translation on Q := MZ such that
E0UE0 = P holds for the conditional expectation defined by Σ0 := ev∗0(S) ⊆ Σ.
An interesting class of examples arises from finite probability spaces M = {1, . . . , n} with
the σ-algebra 2M . Then a Markov kernel P on M is given by the matrix Pij := P (i, {j}) which
is stochastic in the sense that
∑
j Pij = 1 for every i. If µ is a measure on M , encoded in the
row vector p = (pj)1≤j≤n with pj := µ({j}), then µP is represented by the row vector pP . In
particular, the invariance condition νP = ν translates into
∑
i piPij = pj for every j. So P acts
on measures on M (identified with row vectors) by right multiplication and on functions on M
(represented by column vectors) by left matrix multiplication.
If P acts as a symmetric operator on L2(M,ν), then a Markov process (Xn)n∈Z can now be
obtained from the projections Xn : Q = MZ →M and the shift invariant probability measure µ
on Q is determined by
µ(X0 = i) = pi, µ(X1 = i|X0 = j) = Pij . (3.7)
In this context, we have for integers t1 ≤ · · · ≤ tn:∫
Q
f1(ω(t1)) · · · fm(ω(tm)) dµ(ω) =
〈
Mf1P
t2−t1 · · ·Mfm−1P
tm−tm−1Mfm1, 1
〉
L2(M,ν)
=
∫
M
Mf1P
t2−t1 · · ·Mfm−1P
tm−tm−1fm dν(m)
(Theorem 3.22). Applying this relation to δ-functions fj = δij , we obtain
µ(Xt1 = i1, . . . , Xtn = in) = pi1
(
P t2−t1
)
i1i2
(
P t3−t2
)
i2i3
·
(
P tn−tn−1
)
in−1in
.
Reflection Positive Stochastic Processes Indexed by Lie Groups 25
For n = 1, we obtain in particular
µ(Xt = i) = pi = µ(X0 = i).
For n = 2 and t ∈ N0, we obtain
µ(X0 = i,Xt = j) = pi
(
P t
)
ij
.
For t = 1, this leads to
µ(X0 = i, X1 = j) = piPij and µ(X1 = j |X0 = i) = Pij ,
which is (3.7).
4 Measures on path spaces for topological groups
In this section we build a bridge between Markov processes and the reconstruction results from
the preceding section. To this end, we return to the classical setting, where the symmetric Lie
group is (R,R+,− id), but the measure space is the path space P (Q) = QR of a polish topological
group Q.
If ν is a measure on a locally compact group Q and (µt)t≥0 is a convolution semigroup of
symmetric probability measures on Q satisfying ν∗µt = ν for every t > 0, then the Klein–Landau
reconstruction theorem applies. Since the measure ν need not be finite, ν may in particular
be the right invariant Haar measure µQ on Q. The operators Ptf = f ∗ µt define a positive
semigroup structure on L2(Q, ν), and the reconstruction process leads to a reflection positive
unitary representation of R on L2(P (Q),BR, ν) by translation (Theorem 4.9). If µt = γt is
the gaussian semigroup on Rd, this construction leads to an infinite measure on P (Rd) which
realizes in particular the dilation of the heat semigroup as a subrepresentation (cf. [50, 54]).
Here an interesting point is that the map Q × P∗(Q) → P (Q), (q, ω) 7→ qω yields a product
decomposition of P (Q) into the subgroup of constant paths and the subgroup P∗(Q) of pinned
paths. The measures constructed on P (Q) are actually product measures in these coordinates,
where Q carries the measure ν and the measure on P∗(Q) is a probability measure determined
by the µt. There is a natural measure preserving R-action on P∗(Q) by
(Vtω)(s) := ω(−t)−1ω(s− t).
The measure in P∗(Q) defines a Q-valued stochastic process (Xt)t∈R with X1 = 1 (the constant
function). For the special case, where µt = γt is the gaussian semigroup on Rd, this construction
leads to the Wiener measure on P∗(Rd). Presently, we do not know how to obtain a similar
factorization if (R,R+,− idR) is replaced by some (G,S, τ),4 so we discuss the one-dimensional
case in Subsection 4.1.
4.1 One-parameter convolution semigroups of measures on polish groups
In this subsection Q is a polish topological group. We write B for the σ-algebra of Borel subsets
of Q. A kernel K : Q×B→ [0,∞] on (Q,B) (cf. Appendix B) is said to be left invariant if
K(gh, gA) = K(h,A) for all g, h ∈ Q,A ∈ B.
Then µ(A) := K(1, A) is a positive Borel measure on Q such that K(g,A) = µ(g−1A). If,
conversely, µ is a σ-finite Borel measure on Q, then K(g,A) := µ(g−1A) is a left invariant
4This is caused by the ambiguities related to the discrepancy between finite subsets {g1, . . . , gn} of G and
increasing tuples (g1, . . . , gn) with respect to ≺S .
26 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
kernel on Q ×B because, for every A ∈ B, the function g 7→ µ(g−1A) is measurable. In fact,
à := {(x, y) ∈ Q2 : xy ∈ A} is a Borel subset of Q × Q, and à ∩ ({g} × Q) = {g} × g−1A.
We thus obtain a one-to-one correspondence between left invariant Markov kernels K on Q
and Borel probability measures µ on Q. Since the product of two kernels corresponds to the
convolution product of the corresponding measures, this leads to a one-to-one correspondence
between convolution semigroups (µt)t≥0 of Borel probability measures on Q and left invariant
Markov semigroups (Pt)t≥0 on (Q,B). We recall from Remark 3.24(b) that the kernel Pt acts
on functions by right convolution
Ptf = f ∗ µt for t ≥ 0,
where f ∗ µt(x) =
∫
G f(xg) dµt(g).
Consider the path group P (Q) = QR and the subgroup
P∗(Q) := {ω ∈ P (Q) : ω(0) = 1}.
Starting with a convolution semigroup (µt)t≥0, we obtain for the initial distribution δ1 the
Markov process (Xt)t≥0 with independent stationary increments [7, Corollary 35.4]. The distri-
bution of this process is a probability measure Pµ,+ on
P (Q)+ =
{
ω ∈ QR : (∀ t ≤ 0) ω(t) = 1
} ∼= QR×+ .
For the finite distributions on Qn, described by the kernels Pt, t = (t1, . . . , tn), t1 < · · · < tn,
we obtain with Remark B.3
Pt(B) =
∫
Qn
χB(x1, . . . , xn)µt1(dx1)µt2−t1(x−1
1 dx2) · · ·µtn−tn−1(x−1
n−1dxn)
=
∫
Qn
χB(x1, x1x2, . . . , x1 · · ·xn)µt1(dx1)µt2−t1(dx2) · · ·µtn−tn−1(dxn). (4.1)
This means that
Pt = (ψn)∗(µt1 ⊗ µt2−t1 ⊗ · · · ⊗ µtn−tn−1)
for the map
ψn(g1, . . . , gn) = (g1, g1g2, . . . , g1 · · · gn). (4.2)
Remark 4.1. Since the inverse of the map ψn in (4.2) is given by
ψ−1
n (h1, . . . , hn) =
(
h1, h
−1
1 h2, · · · , h−1
n−1hn
)
,
it follows that the random variables
Xt1 , X
−1
t1
Xt2 , . . . , X
−1
tn−1
Xtn
are independent. Since the distribution of X−1
t Xt+s is µs for every t, s ≥ 0, it follows that
(Xt)t≥0 has independent stationary increments (cf. also [7, Corollary 35.4]).
For a curve ω : R → Q, we define (θω)(t) := ω(−t). We write P (Q)± for the subgroup
consisting of paths whose value is 1 on R−, resp., R+ and Q ⊆ P (Q) with the subgroup of
constant paths. For the following proposition, we recall that the involution on the space M(Q)
of finite real Borel measures on Q is defined by∫
Q
f(g) dµ∗(g) =
∫
Q
f
(
g−1
)
dµ(g).
Reflection Positive Stochastic Processes Indexed by Lie Groups 27
Proposition 4.2. Let Pµ,− := θ∗P
µ,+ and Pµ be the image of the measure Pµ,−⊗Pµ,+ on P∗(Q)
under the bijective product map
Φ: P (Q)− × P (Q)+ → P∗(Q), (ω−, ω+) 7→ ω− · ω+.
If µ∗t = µt holds for every t ≥ 0, then this measure is invariant under θ and the one-parameter
group of Borel isomorphisms, given by
(Vtω)(s) := ω(−t)−1ω(s− t), t ∈ R.
Proof. The θ-invariance of P follows immediately from its construction and
θΦ(ω−, ω+) = θ(ω− · ω+) = θ(ω−) · θ(ω+) = θ(ω+) · θ(ω−) = Φ(θ(ω+), θ(ω−)).
We also note that θVtθ = V−t follows from
(θVtθω)(s) = (Vtθω)(−s) = (θω)(−t)−1(θω)(−s− t) = ω(t)−1ω(s+ t) = (V−tω)(s). (4.3)
Since P is uniquely determined by its images under evaluation in finite tuples t = (t1, . . . , tn),
ti < ti+1, it suffices to show that the corresponding distributions Pt on Qn satisfy VtPt =
P(t1+t,...,tn+t). In view of (4.3), we may assume that t > 0. We may further assume that tk = 0
for some k and that tm = −t for some m < k.
First we note that, for measurable functions f1, . . . , fk−1, fk+1, . . . , fn on Q, we have∫
P∗(Q)
∏
j 6=k
fj(ω(tj)) dP
µ(ω)
=
∫
P (Q)+
∫
P (Q)+
k−1∏
j=1
fj(ω−(−tj))
n∏
j=k+1
fj(ω+(tj)) dP
µ,−(ω−)dPµ,+(ω+)
(4.1)
=
∫
Qn−1
f1(gk−1 · · · g1) · · · fk−1(gk−1)fk+1(gk+1) · · · fn(gk+1 · · · gn)
× dµt2−t1(g1) · · · dµtk−1−tk−2
(gk−2)dµ−tk−1
(gk−1)
× dµtk+1
(gk+1)dµtk+2−tk+1
(gk+2) · · · dµtn−tn−1(gn).
For F (ω) =
n∏
j=1
fj(ω(tj)), this leads with tm = −t and tk = 0 to
∫
P (Q)
(VtF )(ω) dPµ(ω) =
∫
P (Q)
F
(
ω(t)−1ω(·+ t)
)
dPµ(ω)
=
∫
P (Q)+
∫
Qm−1
f1
(
ω(t)−1gm−1 · · · g1
)
· · · fm−1
(
ω(t)−1gm−1
)
fm
(
ω(t)−1
)
× fm+1
(
ω(t)−1ω(t+ tm+1) · · · fn
(
ω(t)−1ω(t+ tn)
))
× dµ−tm−1−t(gm−1)dµtm−1−tm−2(gm−2) · · · dµt2−t1(g1)dPµ,+(ω)
=
∫
Qn−1
f1
(
(gm · · · gk−1)−1gm−1 · · · g1
)
· · · fm−1
(
(gm · · · gk−1)−1gm−1
)
fm
(
(gm · · · gk−1)−1
)
× fm+1
(
(gm · · · gk−1)−1gm
)
· · · fk−1((gm · · · gk−1)−1gm · · · gk−2)
× fk+1(gk+1) · · · fn(gk+1 · · · gn) dµ−tm−1−t(gm−1)dµtm−1−tm−2(gm−2) · · · dµt2−t1(g1)
× dµtm+1+t(gm)dµtm+2−tm+1(gm+1) · · ·
× dµtk−tk−1
(gk−1)dµtk+1−tk(gk+1) · · · dµtn−tn−1(gn)
28 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
=
∫
Qn−1
f1
(
g−1
k−1 · · · g
−1
m gm−1 · · · g1
)
· · · fm−1
(
g−1
k−1 · · · g
−1
m gm−1
)
fm
(
g−1
k−1 · · · g
−1
m
)
× fm+1
(
g−1
k−1 · · · g
−1
m+1
)
· · · fk−1
(
g−1
k−1
)
fk+1(gk+1) · · · fn(gk+1 · · · gn)
× dµt2−t1(g1) · · · dµtm−1−tm−2(gm−2)dµtm−tm−1(gm−1)dµtm+1−tm(gm)
× dµtm+2−tm+1(gm+1) · · · dµtk−tk−1
(gk−1)dµtk+1−tk(gk+1) · · · dµtn−tn−1(gn)
(†)
=
∫
Qn−1
f1(gk−1 · · · g1) · · · fm−1(gk−1 · · · gm−1)fm(gk−1 · · · gm)
× fm+1(gk−1 · · · gm+1) · · · fk−1(gk−1)fk+1(gk+1) · · · fn(gk+1 · · · gn)
× dµt2−t1(g1) · · · dµtm−tm−1(gm−1)dµtm+1−tm(gm)dµtm+2−tm+1(gm+1) · · ·
× dµtk−tk−1
(gk−1)dµtk+1−tk(gk+1) · · · dµtn−tn−1(gn)
=
∫
Qn−1
f1(gk−1 · · · g1) · · · fk−1(gk−1)fk+1(gk+1) · · · fn(gk+1 · · · gn)
× dµt2−t1(g1) · · · dµtk−tk−1
(gk−1)dµtk+1−tk(gk+1) · · · dµtn−tn−1(gn).
Here we have used in (†) the symmetry of the measures µt to get rid of the inverses. The
preceding calculation shows that the transformations (Vt)t∈R on P∗(Q) leave the probability
measure Pµ invariant. �
Remark 4.3. Note that the transformations (Vt)t∈R on P∗(Q) satisfy
VtXs = X−1
t Xs+t because (VtXs)(ω) = (V−tω)(s) = ω(t)−1ω(s+ t).
On functions on P∗(Q), we obtain the action
(VtF )(ω) = F (ω(t)−1ω(·+ t))
which is isometric on L2(P∗(Q),BR× , Pµ).
Example 4.4. (a) (Poisson semigroups) For every g ∈ K, we obtain a convolution semigroup
of measures by
µt = e−t
∞∑
k=0
tk
k!
δgk = et(δg−δ0) for t ≥ 0.
This semigroup consists of symmetric measures if and only if δg is symmetric, i.e., g = g−1. In
this case g2n = 1 and g2n+1 = g for all n, so that
µt =
1 + e−2t
2
δ1 +
1− e−2t
2
δg.
Note that the limit for t → ∞ is the measure 1
2(δ1 + δg). This particular example can also be
considered as a one-parameter subsemigroup of the semigroup of Markov matrices on R2, resp.,
Markov operators on the two-element set, where g is the transposition of the two elements (see
also Example 3.10).
(b) (Velocity processes) If K is a Lie group and X ∈ g, then we obtain a semigroup of
measures by
µt = δexp tX for t ≥ 0.
For these semigroups the condition µ∗t = µt for every t ≥ 0 is satisfied only if X = 0, i.e., µt = δ1
for every t.
Reflection Positive Stochastic Processes Indexed by Lie Groups 29
Example 4.5. The class of polish groups is quite large.
(a) Among locally compact groups, the second countable ones carry a complete left invariant
metric, turning them in a polish group. This covers in particular all finite-dimensional Lie
groups with at most countably many connected components.
(b) Separable Fréchet–Lie groups are also polish, and this class contains in particular all
connected Banach–Lie groups whose Lie algebra is separable, gauge groups and groups of
diffeomorphisms of compact manifolds.
(c) Another important example of a polish group is the unitary group U(H)s of a separable
Hilbert space H, endowed with the strong operator topology (cf. [47, Proposition II.1] or [62,
Theorem II.1, p. 93]).
Remark 4.6. In recent years, many interesting one-parameter semigroups of measures have
been studied on infinite-dimensional groups.
(a) On the group Diff(S1) of diffeomorphisms of the circle with respect to the H3/2-metric,
Brownian motion has been constructed by P. Malliavin. A different approach also exhibiting
the invariance under rigid rotations is described in [18] (cf. also [21, 22]).
(b) In [10] Brownian motion on compact groups is studied. This is defined to be a stochastic
process (Xt)t≥0 with values inK withX0 = 1, independent stationary increments and continuous
sample paths. In addition, it is assumed to be symmetric, biinvariant and non-degenerate, i.e., Xt
visits every open subset with positive probability. For the corresponding convolution semigroup
(µt)t≥0 this implies that
lim
t→0
t−1µt
(
V c
)
= 0 (4.4)
for every open 1-neighborhood V in K. Conversely, [10, Theorem 1.2] characterizes the convo-
lution semigroups (µt)t≥0 corresponding to Brownian motions as those which satisfy, in addition
to (4.4), that µt → δ1 weakly on C(K), µ∗t = µt, µt is conjugation invariant (also called central),
and supp(µt) = K for every t > 0.
(c) In [17], Driver studies Brownian motion on the infinite-dimensional Banach–Lie group
W (K) := C∗([0, 1],K) = {ω ∈ C([0, 1],K) : ω(0) = 1},
where K is a connected Lie group with compact Lie algebra, i.e., K ∼= C×Rd for some d ∈ N0 and
a compact Lie group C. Here one has to construct a semigroup (µt)t≥0 of heat kernel measures on
the space W (Rd). On W (Rd) one has to get hold of the smoothness properties of the heat kernel
measures µt corresponding to the Wiener measure. The corresponding Hilbert space H(g) is the
space of finite energy paths with values in the Lie algebra g and the measures µt on W (K) are
quasi-invariant under left and right multiplication with elements of the corresponding Cameron–
Martin group H(K) [17, Theorem 7.7]. The unitary representations of H(K) by left, resp.,
right multiplications has been identified recently with the so-called energy representation of this
group [1]. With similar techniques, the existence of heat kernel measures is also obtained for
the space of pinned paths with values in a compact Riemannian manifold M .
4.2 Standard path space structures for locally compact groups
In this section we assume that the group Q is locally compact and that the convolution semigroup
(µt)t≥0 of probability measures on Q is strongly continuous in the sense that lim
t→0
µt = δ1 = µ0
weakly on the space Cb(Q) of bounded continuous functions on Q. We further assume that ν is
a measure on Q satisfying ν ∗ µt = ν for every t > 0, and, in addition, that the operators
Ptf := f ∗ µt
on L2(Q, ν) are symmetric. If ν is a right Haar measure, then the symmetry of the operators Pt
is equivalent to µ∗t = µt.
30 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Remark 4.7. Let µQ be a right Haar measure on Q and assume that all the measures µt are
symmetric. Integrating the right regular representation (πr(g)f)(x) = f(xg) of Q on L2(Q,µQ),
we obtain a ∗-representation of the convolution algebra M(Q) on L2(Q) by
πr(µ) :=
∫
Q
πr(g) dµ(g),
(
πr(µ)f
)
(x) =
∫
Q
f(xg) dµ(g) = (f ∗ µ)(x).
Then Pt := (πr(µt))t≥0 is a strongly continuous semigroup of hermitian contractions on L2(Q)
(here we use µ∗t = µt) which are Markov operators. This is a positive semigroup structure
because the continuity in measure (SPS4) follows from the continuity of the action of Q (cf.
Remark 3.26). Hence the Klein–Landau reconstruction Theorem 3.22 provides a path space
model for the corresponding dilation representation of R [50].
For Pt as above, formula (3.5)∫
P (Q)
ft1 ⊗ · · · ⊗ ftn dP
µ
t =
∫
Q
f1Pt2−t1f2 · · ·Ptn−tn−1fn dν
in Remark 3.23 specializes for t1 ≤ · · · ≤ tn and 0 ≤ fj to∫
P (Q)
f1 ⊗ · · · ⊗ fn dPµt
=
∫
Qn
f1(g1)f2(g1g2) · · · fn(g1 · · · gn) dν(g1)dµt2−t1(g2) · · · dµtn−tn−1(gn)
=
∫
Qn
f1(g1)f2(g2) · · · fn(gn) dψ∗(ν ⊗ µt2−t1 ⊗ · · · ⊗ µtn−tn−1),
where ψ(g1, . . . , gn) = (g1, g1g2, . . . , g1 · · · gn). We conclude that
Pµt = ψ∗(ν ⊗ µt2−t1 ⊗ · · · ⊗ µtn−tn−1).
Lemma 4.8. Let ν be a measure on Q satisfying ν ∗ µt = ν for every t > 0. Then we obtain
for t1 ≤ · · · ≤ tn and t := (t1, . . . , tn) on Qn a consistent family of measures
Pµt := (ψn)∗(ν ⊗ µt2−t1 ⊗ · · · ⊗ µtn−tn−1).
If Q is a polish group, this leads to a unique measure Pµ on P (Q) with (evt)∗P
µ = Pµt for
t1 < . . . < tn.
Proof. This follows from Remark B.3. For the sake of clarity, we give a direct argument for
the consistency of the measures Pµt . Pick j ∈ {1, . . . , n} and let p : Qn → Qn−1 denote the
projection omitting the jth component. Then
(p ◦ ψn)(g1, . . . , gn) = ψn−1(g1, . . . , gj−1, gjgj+1, gj+2, . . . , gn)
implies for j ≥ 2 that
p∗(ψn)∗(ν ⊗ µt2−t1 ⊗ · · · ⊗ µtn−tn−1)
= (ψn−1)∗(ν ⊗ · · · ⊗ µtj−tj−1 ∗ µtj+1−tj ⊗ · · · ⊗ µtn−tn−1)
= (ψn−1)∗(µt1 ⊗ · · · ⊗ µtj+1−tj−1 ⊗ · · · ⊗ µtn−tn−1).
For j = 1, we obtain
p∗(ψn)∗(ν ⊗ µt2−t1 ⊗ · · · ⊗ µtn−tn−1) = (ψn−1)∗(ν ∗ µt2−t1 ⊗ µt3−t2 ⊗ · · · ⊗ µtn−tn−1)
= (ψn−1)∗(ν ⊗ µt3−t2 ⊗ · · · ⊗ · · · ⊗ µtn−tn−1).
This implies consistency of the measures Pµt on Qn. The consistency condition implies the
existence of a measure Pµ on QR with (evt)∗µ = Pµt for t1 ≤ · · · ≤ tn (cf. Definition 3.11). �
Reflection Positive Stochastic Processes Indexed by Lie Groups 31
From the Klein–Landau reconstruction theorem we immediately obtain the following special-
ization. We refer to [50] for other constructions of this dilation.
Theorem 4.9. Suppose that Q is a second countable locally compact group. Let µ be the measure
on QR corresponding to the symmetric convolution semigroup (µt)t≥0 of probability measures
on Q and the measure ν on Q for which the operators Ptf = f ∗ µt define a positive semigroup
structure on L2(Q, ν). Then the translation action (Utω)(s) := ω(s − t) on P (Q) = QR is
measure preserving and µ is invariant under (θω)(t) := ω(−t). We thus obtain a reflection
positive one-parameter group of Markov type on E := L2(P (Q),BR, µ) with respect to E+ :=
L2(P (Q),BR+ , µ), for which E0 := ev∗0(L2(Q, ν)) ∼= L2(Q, ν) and Ê ∼= L2(Q, ν) with q(F ) = E0F
for F ∈ E+. We further have
E0UtE0 = Pt holds for Ptf = f ∗ µt,
so that the U -cyclic subrepresentation generated by E0 is a unitary dilation of the hermitian
one-parameter semigroup (Pt)t≥0 on L2(Q, ν).
Example 4.10. (a) For Q = Rd, the heat semigroup is given on L2(Rd) by
et∆f = f ∗ γt where dγt(x) =
1
(2πt)d/2
e−
1
2
‖x‖2
t dx.
We call the corresponding measure on QR the Lebesgue–Wiener measure (cf. Theorem 4.9).
(b) If Q is a finite-dimensional Lie group and X1, . . . , Xn is a basis of the Lie algebra, then we
obtain a left invariant Laplacian by ∆ :=
n∑
j=1
L2
Xj
, where LXj denotes the right invariant vector
field with LXj (1) = Xj . Then there also exists a semigroup (µt)t≥0 of probability measures
on Q such that [53, Section 8]
et∆f = f ∗ µt for t ≥ 0.
Accordingly, we obtain a Haar–Wiener measure on the path space QR.
5 Gaussian (G,S, τ )-probability spaces
In this section we discuss the second quantization functor and its connection to gaussian (G,S, τ)-
probability spaces. We then discuss equivalence of gaussian measures for reproducing kernel
Hilbert spaces. The main results of this section are contained in Subsection 5.4, where we
discuss gaussian measures on the space of distributions on a Lie group. Here the distribution
vectors of unitary representations play an important role.
5.1 Second quantization and gaussian processes
Definition 5.1. Let H be a real Hilbert space. A gaussian random process indexed by H is
a random process (ϕ(v))v∈H on a probability space (Q,Σ, µ) such that
(GP1) {ϕ(v) : v ∈ H} is full, i.e., these random variables generate the σ-algebra Σ modulo zero
sets.
(GP2) Each ϕ(v) is a gaussian random variable of mean zero.
(GP3) 〈ϕ(v), ϕ(w)〉 = 〈v, w〉 is the inner product on H.
32 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
According to [64, Theorems I.6, I.9], gaussian random processes indexed by H exist and are
unique up to isomorphisms of probability spaces. This means that, if (Q,Σ, µ) is the corre-
sponding probability space, then the algebra L∞(Q,µ) with its state given by µ is uniquely
determined by H (cf. [64, Section 1.1]). Its projections correspond to Σ/Jµ, where Jµ is the
ideal of zero sets5.
Theorem 5.2. Let V be a real vector space, V ∗ be its algebraic dual and B∗ be the smallest σ-
algebra for which all evaluation functions V ∗ → V, α 7→ α(v), are measurable. Then a function
χ : V → C is the Fourier transform χ = µ̂ of a measure µ on (V ∗,B∗) if and only if χ is
positive definite and continuous on every finite-dimensional subspace. In this case µ is uniquely
determined.
Proof. Since the Fourier transform µ̂ is positive definite and sequentially continuous, it is in
particular continuous on every finite-dimensional subspace of V . The converse follows from [68,
Theorem 16.2]. �
Example 5.3. Theorem 5.2 implies that the gaussian process indexed by H may be realized by
the probability measure γH on (Ha,B∗) (the algebraic dual) whose Fourier transform is
γ̂H(v) = e−
‖v‖2
2 .
Here ϕ(v)(α) = α(v), and∫
Ha
eiα(v) dγH(α) = e−
‖v‖2
2 implies
∫
Ha
α(v)2 dγH(α) = ‖v‖2.
This leads to〈
eiϕ(v), eiϕ(w)
〉
= e−
‖v−w‖2
2 = e−
‖v‖2
2 e−
‖w‖2
2 e〈v,w〉,
so that Kv := eiϕ(v)e
‖v‖2
2 satisfies
〈Kv,Kw〉 = e〈v,w〉.
Example 5.4. If H is finite-dimensional, so that H ∼= Rd for some d ∈ N0, then Ha ∼= Rd and
dγH(x) = 1
(2π)d/2
e−
‖x‖2
2 dx is a gaussian measure.
We now introduce the second quantization. Conceptually the easiest way to define second
quantization as a functor is to associate to a (real or complex) Hilbert space H its Fock space
F(H) :=
⊕̂
n∈N0
Sn(H),
where Sn(H) := (H⊗n)Sn is the closed subspace of Sn-invariant vectors in the n-fold tensor
power of H. For v1, . . . , vn ∈ H, we define the symmetric product by
v1 ∨ · · · ∨ vn := P+(v1 ⊗ · · · ⊗ vn),
5A short proof for the uniqueness can be derived from reproducing kernel techniques. If ϕj : H → L2(Qj , µj),
j = 1, 2, are two realizations, then the corresponding covariance kernels on H coincide, so that there exists
a unique unitary operator Φ: L2(Q1, µ1) → L2(Q2, µ2) such that Φ ◦ ϕ1 = ϕ2. Accordingly, Φ ◦ A1 ◦ Φ−1 = A2
for the von Neumann algebras Aj := {eiϕj(v) : v ∈ H}′′ ∼= L∞(Qj , µj).
Reflection Positive Stochastic Processes Indexed by Lie Groups 33
where P+ : H⊗n → Sn(H) is the orthogonal projection. The inner products of such elements are
given by
〈v1 ∨ · · · ∨ vn, w1 ∨ · · · ∨ wn〉 = 〈v1 ∨ · · · ∨ vn, w1 ⊗ · · · ⊗ wn〉
=
1
n!
∑
σ∈Sn
〈vσ(1), w1〉 · · · 〈vσ(n), wm〉.
In particular, with vn = v ∨ · · · ∨ v︸ ︷︷ ︸
n-times
, we have
〈vn, wn〉 = 〈v, w〉n and ‖vn‖ = ‖v‖n. (5.1)
Clearly, every contraction A : H → K defines a contraction Γ(A) : F(H)→ F(K) by
Γ(A)(v1 ∨ · · · ∨ vn) := Av1 ∨ · · · ∨Avn
and it is clear that Γ(AB) = Γ(A)Γ(B) and Γ(A∗) = Γ(A)∗. In particular, we obtain a represen-
tation of the involutive semigroup of contractions on H in the Fock space F(H) and F defines
an endofunctor from the category of Hilbert spaces whose morphism are contractions into itself.
The problem with this approach is that it completely ignores positivity issues.
If H is a Hilbert space, then any v ∈ H defines a function 〈v, ·〉, u 7→ 〈v, u〉, which is linear if
H is real and antilinear if H is complex. For
Exp(v) :=
∞∑
n=0
1√
n!
vn,
we now derive from (5.1) that Exp(v) ∈ F(H) and
〈Exp(v),Exp(w)〉 =
∞∑
n=0
1
n!
〈v, w〉n = e〈v,w〉.
This leads to an embedding
Φ: F(H)→ CH, Φ(ξ)(v) := 〈ξ,Exp(v)〉,
where
Φ(v1 ∨ · · · ∨ vn)(v) =
1√
n!
〈v1 ∨ · · · ∨ vn, vn〉 =
1√
n!
n∏
j=1
〈vj , v〉.
The image of Φ is the reproducing kernel space FRK(H) with kernel
K(v, w) = 〈Exp(w),Exp(v)〉 = e〈w,v〉.
We may thus identify F(H) with the reproducing kernel Hilbert space FRK(H).
For a contraction A : H → K, we have
Φ(Av1 ∨ · · · ∨Avn)(v) =
1√
n!
〈Av1 ∨ · · · ∨Avn, vn〉 =
1√
n!
〈v1 ∨ · · · ∨ vn, (A∗v)n〉,
so that the operator Γ(A) acts on the reproducing kernel space FRK(H)→ FRK(K) simply by
(Γ(A)F )(v) := F (A∗v), ‖A‖ ≤ 1, F ∈ HK ⊆ CH, v ∈ K.
34 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
5.2 Application to reflection positive representations
Typical examples of gaussian (G,S, τ)-probability spaces arise as follows. LetH be a real Hilbert
space and (Q,Σ, γH) be a realization of the gaussian random process (ϕ(v))v∈H indexed by H
(Definition 5.1). Using the realization from Example 5.3, where Q = Ha is the algebraic dual
space, we obtain an action of the orthogonal group O(H) on (Q,Σ, µ) by measure preserving
automorphisms.
For the following proposition, we recall the concept of a reflection positive representation of
(G,S, τ) from Definition 2.6.
Lemma 5.5. If π : S → B(H) be is a continuous ∗-representation of the topological involu-
tive semigroup (S, ]) by contractions, then we obtain on Γ(H) ∼= L2(Ha, γH) by Ps := Γ(π(s))
a standard positive semigroup structure on the probability space (Ha, γH).
Proof. Here we use that
ε2γH({|Psf − Ps0f | ≥ ε}) ≤
∫
Ha
|Psf − Ps0f |2 dγH → 0
for s → s0 follows from the continuity of the representation Γ ◦ π of S on Γ(H) (cf. Lem-
ma A.1). �
Proposition 5.6. Let (U, E , E+, θ) be a reflection positive orthogonal representation of (G,S, τ)
for which E0 is U -cyclic and E+ is generated by (UsE0)s∈S. Then second quantization leads to
a (G,S, τ)-probability space ((QH,Σ, γH),Σ0,Γ(U),Γ(θ)), where Σ0 ⊆ Σ is the smallest σ-algebra
for which the functions (ϕ(v))v∈E0 are measurable.
Proof. (GP1-4) Clearly, every Γ(Ug) and Γ(θ) are automorphisms of the algebra L∞(QH,Σ, γH)
satisfying
Γ(θ)Γ(Ug)Γ(θ) = Γ(θUgθ) = Γ(Uτ(g)) and θE0 = E0.
The continuity of the unitary representation (Γ(Ug))g∈G on L2(QH, γH) implies the continuity
in measure of the G-action on L∞(QH, γH) (Lemma 5.5).
(GP5) Our definition of Σ0 implies that Σ+ is the smallest σ-algebra for which the functions
(ϕ(v))v∈E+ are measurable and since E0 is U -cyclic in E , (GP5) is also satisfied.
Reflection positivity of the representation Γ(U) of (G,S, τ) follows from [50, Remark 3.8] and
Γ(E+) = L2(QH,Σ+, γH). �
5.3 Equivalence of gaussians measures for reproducing kernel Hilbert spaces
Let X be a set and E = C[X] the free complex vector spaces over X. Then positive definite
kernels K on X are in one-to-one correspondence with positive semidefinite hermitian forms
on E. Any such kernel defines a Hilbert subspace HK ⊆ E∗ with continuous point evaluations.
More generally, we may consider for a real locally convex space E continuous bilinear hermi-
tian kernels K : E×E → C and the corresponding subspaces of the topological dual space E′ [61].
Such a kernel is positive definite if and only if the canonical sesquilinear extension to the com-
plexification EC is a positive semidefinite hermitian form. Suppose that E is nuclear. Then, for
any such K, the function
ϕK(v) := e−
1
2
K(v,v)
on E is continuous and positive definite, hence is the Fourier transform of a uniquely determined
gaussian measure γK on E′. We want to express conditions on pairs of kernels K and Q which
characterize the equivalence of the measures γK and γQ on E′.
Reflection Positive Stochastic Processes Indexed by Lie Groups 35
According to [38], two gaussian measures are either mutually singular of equivalent. The
following theorem is a reformulation of [68, Theorem 10.1] (cf. also [37, Theorem 4.1/4.2] or [11,
Corollary 6.4.11]).
Theorem 5.7. Let E be a nuclear real locally convex space. For two continuous positive
semidef inite hermitian forms K and Q on E, the corresponding gaussian measures γK and γQ
on E′ are equivalent if and only if HK = HQ and there exists an operator T ∈ GL(HK) for
which TT ∗ − 1 is Hilbert–Schmidt and
Q(x, y) = 〈TKy, TKx〉 for x, y ∈ E.
Remark 5.8. (a) That the gaussian measure γK on E′ determines the Hilbert subspaceHK ⊆ E′
follows from [68, Theorem 9.1] which asserts thatHK consists precisely of those linear functionals
α ∈ E′ for which γK is quasi-invariant under the translation τα(β) := α+ β. As a consequence,
the equivalence of γK and γQ implies HK = HQ.
(b) Recall from [48, Theorem I.2.8] that HK = HQ is equivalent to the existence of positive
constants c1, c2 > 0 such that the kernels
K − c1Q and c2Q−K
are positive definite. If K and Q are real-valued, this is equivalent to
c1Q(x, x) ≤ K(x, x) ≤ c2Q(x, x) for x ∈ E.
This in turn is equivalent to the existence of a bounded invertible positive operator A ∈ B(HK)
such that
Q(x, y) = KA(x, y) := 〈AKy,Kx〉 for x, y ∈ E.
In view of [48, Corollary I.2.6], we have A = TT ∗, where T : HQ → HK , f 7→ f is the identity.
This implies that
T ∗(Kx) = Qx for x ∈ E.
In particular, we have
Q(x, y) = 〈Qx, Qy〉HQ = 〈TT ∗Kx,Ky〉HK .
We conclude that, for f ∈ HQ = HK , we have
‖f‖2HK = ‖Tf‖2HK = 〈T ∗Tf, f〉HQ .
Therefore the equivalence of the corresponding gaussian measure is equivalent to T ∗T −1 being
Hilbert–Schmidt.
Remark 5.9. Let E be a real vector space and endow it with the finest locally convex topology
for which all seminorms on E are continuous. Then E is nuclear if and only if E is of at most
countable dimension [65, Proposition 50.1, Theorem 51.2]. In any case, its topological dual space
is E′ = E∗ because every linear functional on E is continuous.
Regardless of the nuclearity of E, every positive definite function ϕ : E → C which is con-
tinuous on all finite-dimensional subspace is the Fourier transform of a B∗-measure on E∗
(Theorem 5.2). This applies in particular to all functions of the form ϕ(v) := e−
1
2
K(v,v), where
K : E × E → R is a positive semidefinite symmetric bilinear form on E.
36 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
5.4 Gaussian measures on distributions on Lie groups
If G is a Lie group, then D(G) := C∞c (G) is an involutive algebra with respect to the convo-
lution product and the involution ϕ∗(g) := ϕ(g−1)∆G(g−1), where ∆G is the modular function
satisfying
∆G(y)
∫
G
f(xy) dµG(x) =
∫
G
f(x) dµG(x) for f ∈ Cc(G), y ∈ G.
Accordingly, we call a distribution D ∈ D′(G) (the space of antilinear continuous functionals
on D(G)) positive def inite, if it is a positive functional on this algebra, i.e.,
D(ϕ∗ ∗ ϕ) ≥ 0 for ϕ ∈ D(G).
Since D(G) is nuclear, every positive definite distribution D ∈ D′(G) determines a gaussian
measure γD on D′(G).
For a unitary representation (π,H) of G we write H−∞ for the space of continuous antilinear
functionals on H∞, the space of distribution vectors, and note that we have a natural linear
embedding H ↪→ H−∞, v 7→ 〈v, ·〉. Accordingly, we also write 〈α, v〉 = 〈v, α〉 for α(v), α ∈ H−∞
and v ∈ H∞. The group G acts naturally on H−∞ by
(π−∞(g)α)(v) := α
(
π(g)−1v
)
,
so that we obtain a G-equivariant chain of continuous inclusions
H∞ ⊆ H ⊆ H−∞ (5.2)
(cf. [14, Section 8.2]). It is D(G)-equivariant, if we define the representation of D(G) on H−∞
by
(
π−∞(ϕ)α
)
(v) :=
∫
G
ϕ(g)α
(
π(g)−1v
)
dµG(g) = α(π(ϕ∗)v).
Proposition 5.10 ([49, Proposition 2.8]). Let D ∈ D′(G) be a positive definite distribution
on the Lie group G and HD be the corresponding reproducing kernel Hilbert space with kernel
K(ϕ,ψ) := D(ψ∗ ∗ ϕ) obtained by completing D(G) ∗D with respect to the scalar product
〈ψ ∗D,ϕ ∗D〉 = D(ψ∗ ∗ ϕ).
Then the following assertions hold:
(i) HD ⊆ D′(G) and the inclusion γD : HD → D′(G) is continuous.
(ii) We have a unitary representation (πD,HD) of G by
πD(g)E = g∗E, where (g∗E)(ϕ) := E(ϕ ◦ λg)
and the integrated representation of D(G) on HD is given by πD(ϕ)E = ϕ ∗ E.
(iii) There exists a unique distribution vector αD ∈ H−∞D with αD(ϕ ∗D) = D(ϕ) and
π−∞(ϕ)αD = ϕ ∗D for ϕ ∈ D(G).
(iv) γD extends to a D(G)-equivariant injection H−∞D ↪→ D′(G) mapping αD to D.
Reflection Positive Stochastic Processes Indexed by Lie Groups 37
Remark 5.11. We consider a reflection positive distribution vector α ∈ H−∞ for a unitary
representation (π,H) of G. This leads to an embedding
ηα : H → D′(G), ηα(v)(ϕ) := 〈v, π−∞(ϕ)α〉 = 〈π(ϕ∗)v, α〉.
which is injective if and only if α is cyclic. This establishes a one-to-one correspondence between
distribution vectors and G-equivariant continuous linear maps H → D′(G) (Proposition 5.10).
Actually we obtain an equivariant embedding
H−∞ ↪→ D′(G)
by dualizing the linear map D(G) → H, ϕ 7→ π−∞(ϕ)α. This in turn leads to the positive
definite function
S(ϕ) := e−
1
2‖π
−∞(ϕ)α‖2 = e−
1
2D(ϕ∗∗ϕ) for D(ϕ) := α(π−∞(ϕ)α), ϕ ∈ D(G).
We thus obtain a G-invariant gaussian probability measure γD on D′(G) by Minlos’ theorem.
Lemma 5.12. If α ∈ H−∞ is cyclic and (ρ, V ) is a finite-dimensional irreducible representation
of G, then the multiplicity of ρ in H is bounded by dimV . In particular, the V -isotypic subspace
of H is finite-dimensional and dimHG ≤ 1.
Proof. Let n ∈ N and assume that V n is a subrepresentation of H. Then V n is finite-dimen-
sional and generated by a distribution vector which actually must be an element (v1, . . . , vn)
∈ V n. Suppose that λ1, . . . , λn ∈ C satisfy
∑
j λjvj = 0. Then
∑
j λjπ(g)vj = 0 for every g ∈ G,
and hence
∑
j λjwj = 0 for every (w1, . . . , wn) ∈ V n because (v1, . . . , vn) is cyclic in V n. This
leads to λj = 0 for every j, so that the elements v1, . . . , vn are linearly independent. �
Theorem 5.13. Let G be a Lie group and D,E ∈ D′(G) be positive definite distributions.
Then the corresponding gaussian measures γD and γE on D′(G) are equivalent if and only if the
following conditions are satisfied
(i) D can be written as an orthogonal sum D = D0 +
∑
n∈J Dn, where J ⊆ N and the
representation on the subspaces HDn, n ∈ J , are finite-dimensional isotypic and mutually
disjoint.
(ii) E = D0 +
∑
n∈J En with HDn = HEn, and there exist intertwining operators T0 = idHD0
and Tn ∈ BG(HDn) with TnDn = En in H−∞Dn = HDn and
∑
n∈J ‖TnT ∗n − 1‖22 <∞.
Proof. We shall use Theorem 5.7. If γD ∼ γE , then HD = HE ⊆ D′(G) and the identity map
T : HD → HE , f 7→ f is a G-equivariant operator, so that T ∗T ∈ B(HD) is a G-intertwining
operator. The requirement that T ∗T − 1 is Hilbert–Schmidt implies that its range is a sum
of finite-dimensional subrepresentations. In view of the preceding lemma, it can be written as
⊕j∈JHj , where the Hj are isotypic, finite-dimensional and mutually disjoint. Then Tn := T |HDn
maps HDn into itself and
‖TT ∗ − 1‖22 =
∑
n∈J
‖TnT ∗n − 1‖22.
The converse implication follows from Theorem 5.7. �
Corollary 5.14. If D is a positive definite distribution on G, then the following are equivalent
(a) HD contains no G-invariant subspace of finite positive dimension.
38 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
(b) For any two different distribution vectors E,E′ ∈ H−∞D ⊆ D′(G), the corresponding gaus-
sian measures γE and γE′ are inequivalent.
Example 5.15. We discuss the special case G = Rn. According to the Bochner–Schwartz
theorem, a distribution D ∈ D′(G) is positive definite if and only if it is the Fourier transform
D = µ̂ of a tempered measure µ on the dual group Ĝ ∼= Rd, and then HD ∼= L2(Rd, µ) with the
representation
(πD(x)f)(y) = e−ixyf(y).
For D = µ̂ and E = ν̂, the equality of the corresponding Hilbert spaces is equivalent to the
existence of positive constants c1, c2 > 0 with
c1ν ≤ µ ≤ c2ν,
which is equivalent to the equivalence of the measures µ and ν with the additional requirement
that δ := dµ
dν satisfies 0 < c1 ≤ δ ≤ c2 (cf. [48] or [37]). Then L2(Rd, µ) = L2(Rd, ν), and
the identity T : L2(Rd, ν) → L2(Rd, µ) is an isomorphism of Banach spaces. We then have
T ∗(f) = δf and the equivalence of the gaussian measures is equivalent to the multiplication
operator Mδ−1 = T ∗T − 1 being a Hilbert–Schmidt operator on L2(Rd, ν) (Theorem 5.7). This
is equivalent to the condition that the restriction of ν to the subset {δ 6= 1} is atomic, so that the
values of δ in these points are defined, and the Hilbert–Schmidt condition can be expressed as∑
δ(x)6=1
|δ(x)− 1|2 <∞.
We conclude that γD ∼ γE is equivalent to µ = ν on the complement of an at most countable
set S of atoms for both measures, which satisfies∑
s∈S
∣∣∣∣µ({s})
ν({s})
− 1
∣∣∣∣2 <∞.
Remark 5.16. (a) Theorem 5.13 generalizes in an obvious way to continuous positive definite
functions on a topological group G.
(b) Example 5.15 generalizes in the obvious fashion to positive definite functions on a locally
compact abelian group, or, more generally, on a nuclear abelian group (cf. [6]).
The following theorem covers in particular the case of one-dimensional Lie groups.
Theorem 5.17. Let A be a selfadjoint operator on the Hilbert space H. Then the following are
equivalent:
(a) The gaussian measure γH can be realized on H−∞(A), the dual space of
H∞(A) =
⋂
n∈N0
D
(
An
)
.
(b) There exists an N ∈ N such that the bounded operator (1 +A2)−N is Hilbert–Schmidt.
(c) The Fréchet space H∞(A) is nuclear.
Proof. (a) ⇒ (b) Let E ⊆ R be a bounded subset, H1 := P (E)H (for the spectral measure P
of A) and H2 := H⊥1 . Then we accordingly have A = A1⊕A2, where the operator A1 is bounded
and D(A) = H1 ⊕D(A2). This implies that
H∞(A) = H1 ⊕H∞2 (A2) and thus H−∞(A) = H1 ⊕H−∞2 (A2).
Reflection Positive Stochastic Processes Indexed by Lie Groups 39
Therefore γH can be realized on H−∞ if and only if γH1 can be realized on H1, which means
that H1 is finite-dimensional, and γH2 can be realized on H−∞2 (A2).
Therefore (a) implies that all spectral projections P ([a, b]), a < b, have finite-dimensional
range. As a consequence, A is diagonalizable with discrete spectrum and finite-dimensional
eigenspaces; in particular H is separable. Let (λn)n∈N be the eigenvalues of A, counted with
multiplicities and (en)n∈N an ONB of H with Aen = λnen for every n ∈ N. Now
H−∞(A) =
{∑
n
xnen : (∃N ∈ N)
∑
n
(
1 + λ2
n
)−2N |xn|2 <∞
}
is the union of the subspaces
H−2N (A) =
{∑
n
xnen :
∑
n
(
1 + λ2
n
)−2N |xn|2 <∞
}
.
We realize the gaussian measure γH on the product space CN. Then every subspace H−2N (A)
is measurable, and
1 = γH
(
H−∞(A)
)
= lim
N→∞
γH
(
H−2N (A)
)
implies that γH(H−2N (A)) > 0 for some N ∈ N. From the example in [68, p. 153], it now follows
that ∥∥(1 +A2
)−N∥∥2
2
=
∑
n
(
1 + λ2
n
)−2N
<∞
(see also [17, Theorem 5.2]).
(b)⇔ (c): The space D∞(A) :=
⋂
n∈ND(An) coincides with the space D∞(B) for B := 1+A2,
and B has the additional property that Bn ≤ Bn+1 for n ∈ N0. The topology on D∞(B) is
defined by the seminorms pn(v) := ‖Bnv‖, n ∈ N0. In view of [65, Definition 50.1], the nuclearity
of the space H∞ = D∞(B) is equivalent to the condition that, for every n ∈ N, there exists an
m > n, such that the natural map Dm(B)→ Dn(B) is nuclear. Since this map can be identified
with the map Bn−m : H → H, we see that the nuclearity of H∞ is equivalent to the existence
of some nuclear power of B−1, which is equivalent to the existence of some power which is
Hilbert–Schmidt. This means that D∞(A) is nuclear if and only if some operator (1 + A2)−N ,
N ∈ N, is Hilbert–Schmidt.
(c) ⇒ (a) follows from the Bochner–Minlos theorem. �
Example 5.18. (a) In the context of reflection positivity on curved spacetimes, a natural
class of selfadjoint operators arises as follows [31, 32]. We call a connected complete orientable
Riemannian manifold (M, g) a quantizable static space-time if there exists a complete Killing
vector field ξ orthogonal to a hypersurface Σ ⊆M such that all integral curves of ξ intersect Σ
exactly once. Then the flow of ξ induces an isometry Φ: R× Σ→M of Riemannian manifolds
and θ(Φ(t, x)) = Φ(−t, x) is an isometric involution exchanging the two open subsets Ω± =
{Φ(t, x) : ±t > 0, x ∈ Σ}. Let ∆ be the Laplacian of (M, g) and m > 0. Then the free
covariance operator C := (m2 −∆)−1 is a bounded selfadjoint operator on L2(M). For s ∈ R,
we write Hs for the completion of L2(M) w.r.t. the scalar product 〈f, g〉 := 〈f, C−sg〉L2(M) (the
Sobolev spaces) and obtain a Fréchet space
S := D∞
(
m2 −∆
)
= D∞(∆) =
⋂
s∈R
Hs,
40 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
but in general this space is not nuclear because the Laplacian may have continuous spectrum
(which contradicts the statement in [32, Definition 2.1]). But if Σ is compact and
H :=
1
2
+
1
2
(
Q2
t −∆
)
,
where Qt is the multiplication with the global time function, then there exists a power (1+H)−N
which is trace class, so that
S := D∞(H) ⊆ L2(M)
is a nuclear space. The operator H is the sum of the Hamiltonian of the harmonic oscillator in
the time direction and the Laplacian of Σ [4, Proposition 7.5]. We refer to [4, Section 7.4] for
a discussion of the corresponding gaussian measures on the dual space S ′.
(b) For M = Rd and H = L2(Rd), the Hamiltonian of the d-fold harmonic oscillator
H =
1
2
d∑
j=1
Q2
j − ∂2
j , (Qjf)(x) = xjf(x), (∂jf)(x) =
∂
∂xj
f(x)
leads to D∞(H) = S(Rd) which also is a nuclear space.
Corollary 5.19. Let (π,H) be a continuous unitary representation of the finite-dimensional Lie
group G and put ∆ =
∑
j X
2
j for a basis X1, . . . , Xn of g. Then the following are equivalent:
(a) The gaussian measure γH can be realized on H−∞.
(b) There exists an N ∈ N such that (1− dπ(∆))−N is a Hilbert–Schmidt operator.
(c) The Fréchet space H∞ is nuclear.
(d) π is trace class, i.e., for each f ∈ C∞c (G), the operator π(f) is trace class.
Proof. In view of Theorem 5.17, the equivalence of (a)-(c) follows from Nelson’s theorem as-
serting that H∞ = H∞(A) holds for the selfadjoint operator A := dπ(∆) [67, Theorem 4.4.4.5].
For the equivalence with (d) we refer to [15, Proposition 1.11]. �
Proposition 5.20. Let (π,H) be a unitary representation of the Lie group G. Then the
space H∞ is nuclear in the following cases:
(a) If G is compact and π is distribution cyclic.
(b) If G is connected semisimple and π is irreducible.
(c) If G is connected nilpotent and π is irreducible.
Proof. (a) (Sketch) We denote irreducible representation with highest weight λ by (πλ, Vλ).
We also set d(λ) := dimVλ. We have H = ⊕λHλ, where Hλ is the isotypic subspace of highest
weight λ. The multiplicity of the simple G-module Vλ in Hλ is bounded by d(λ) (Lemma 5.12).
We can then choose the basis Xj such that −∆ coincides with the Casimir element of g. Let
ρ = 1
2
∑
β∈∆+ β. Then
−dπ(∆)|Hλ =
(
‖λ+ ρ‖2 − ‖ρ‖2
)
1
by Freudenthal’s lemma [48, Lemma IX.5.2]. We therefore have∥∥(1− dπ(∆))−N
∥∥2
2
=
∑
λ
(
1 + ‖λ+ ρ‖2 − ‖ρ‖2
)−2N
dimHλ
≤
∑
λ
(
1 + ‖λ+ ρ‖2 − ‖ρ‖2
)−2N
d(λ)2.
Reflection Positive Stochastic Processes Indexed by Lie Groups 41
This expression is finite for some N ∈ N because d(λ) is bounded by a polynomial in ‖λ‖ [67,
Lemma 4.4.2.3]. This means that there exists an N ∈ N for which (1 − dπ(∆))−N is Hilbert–
Schmidt. Therefore H∞ = D∞(dπ(∆)) = D∞((1− dπ(∆))N ) is nuclear.
(b) ([13, Theorem 2.1]) Write g = k ⊕ s for the Cartan decomposition of g. Then we can
choose the basis Xj such that ∆ = Ω + 2∆k where Ω is the Casimir element and ∆k is the
Laplacian for k. As Ω acts by a scalar and the dimension of Hλ is bounded by d(λ)2, the claim
follows as in (a).
(c) ([13, Proposition 1.9(a)]) According to [39, Theorem 3, p. 103], we can realize π in
L2(Rn, dx) such that H∞ coincides with the Schwartz space S(Rn) which is nuclear. �
Example 5.21. For the distribution D = δ1 on the compact Lie group G, the corresponding
representation is the regular representation on HD ∼= L2(G). In this case L2(G)∞ = C∞(G) (f.i.
by the Dixmier–Malliavin theorem) is a nuclear space and L2(G)−∞ = C−∞(G) is the space of
distributions on G. In particular, the measure γD can be realized in this space.
Remark 5.22. The condition H∞ = H is equivalent to the smoothness of the representation,
which in turn is equivalent to the boundedness of the operators dπ(X), X ∈ g. If this is the case,
then H−∞ = H. If H is infinite-dimensional it is not nuclear, and since H ⊆ Ha (the algebraic
dual) is a zero set for γH, the corresponding gaussian measure can not be realized on H−∞.
Examples arise from the multiplication representation of an abelian Lie group G on the space
H = L2(Ĝ, µ), where µ is a compactly supported measure on the dual group Ĝ = Hom(G,T).
These representations are smooth and generated by the cyclic vector 1. If H is infinite-
dimensional, then it is not nuclear. In particular, γD is not realized on the subspace H−∞D
of D′(G).
Example 5.23. The multiplication representation of G = R on L2(R, µ) for the measure µ =∑
n∈N
1
2n δ 1
n
is norm continuous and H∞ = H is not nuclear.
5.5 Ergodicity of γH for G
Let (π,H) be an orthogonal representation of G on the real Hilbert space H and γH the cor-
responding gaussian measure. We consider the so-called gaussian action of G on the gaussian
probability space (Ha, γH) on the algebraic dual Ha. The measure γH is G-ergodic if and only if
Γ(H)G = L2(Ha, γH)G = C1.
In this section we derive a criterion for this condition to be satisfied.
Lemma 5.24. Let (πj ,Hj)j=1,2 be two orthogonal representations of the group G.
(i) If H1 has no non-zero finite-dimensional subrepresentations, then all finite-dimensional
invariant subspaces of H1 ⊗H2 are zero. In particular, (H1 ⊗H2)G = {0}.
(ii) The subspace (H1⊗H2)f generated by the finite-dimensional invariant subspaces coincides
with H1,f ⊗H2,f .
Proof. (i) First we show that the subspace (H1 ⊗H2)G of fixed vectors is trivial. Since (H1 ⊗
H2)G can be identified with the space of Hilbert–Schmidt intertwining operators A : H∗2 → H1,
any such operator leads to the self intertwining operator AA∗ ∈ B2(H1) and its eigenspaces are
finite-dimensional G-invariant subspaces, hence trivial. Let F ⊆ H1⊗H2 be a finite-dimensional
invariant subspace. Then idF ∈ F ⊗F∗ ⊆ H1⊗H2⊗H∗1⊗H∗2 is a fixed vector, so that F = {0}
follows from the preceding argument.
(ii) We write
H1 ⊗H2 = (H1,f ⊗H2,f )⊕
(
H⊥1,f ⊗H2
)
⊕
(
H1,f ⊗H⊥2,f
)
42 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
and apply the preceding proposition to see that only the first summand contains non-zero finite-
dimensional invariant subspaces. �
For unitary representations, one can also introduce the terminology from measure preserving
actions on a probability space, where H ∼= L2
0(X,Σ, µ) = {f ∈ L2(X,Σ, µ) :
∫
X f dµ = 0}.
Definition 5.25. Let (π,H) be a unitary representation and (π∗,H∗) the dual representation.
We say that π is:
(a) ergodic if HG = {0},
(b) weakly mixing if π ⊗ π∗ is ergodic,
(c) mixing if G is locally compact and all matrix coefficients are contained in C0(G).
The following proposition is an elaboration of the main result of [63] whose main focus is the
equivalence of (i) and (ii).
Theorem 5.26. (I.E. Segal) For an orthogonal representation (π,H) of the group G, the fol-
lowing are equivalent:
(i) H contains no non-zero finite-dimensional invariant subspaces.
(ii) The gaussian measure γH is G-ergodic.
(iii) π is weakly mixing.
(iv) For every orthogonal representation (ρ,K), the representation π ⊗ ρ is ergodic.
If these conditions are satisfied, then, for every N ∈ N, the product measure γ⊗NH is also ergodic.
Proof. (i) ⇔ (ii): (cf. [8, Proposition A.1.12, Corollary A.7.15])6 If F ⊆ H is a finite-
dimensional invariant subspace, then γF (which is equivalent to Lebesgue measure on F) is
not ergodic. Now γH ∼= γF ⊗ γF⊥ implies that γH is not ergodic.
If, conversely, all finite-dimensional invariant subspaces of H are trivial, then all G-fixed
vectors in Sn(H) ⊆ H⊗n are trivial if n > 0 (Lemma 5.24). Hence the assertion follows from
the G-equivariant decomposition Γ(H) ∼= CΩ⊕
⊕
n>0 S
n(H).
(i) ⇒ (iv) follows from Lemma 5.24(i).
(iv) ⇒ (iii) is trivial.
(iii)⇒ (i): If F ⊆ H is a finite-dimensional invariant subspace, then idF ⊆ F⊗F∗ ∼= H⊗H∗
is a fixed vector.
If (i) is satisfied, then the canonical representation on HN also contains no non-zero finite-
dimensional invariant subspace, so that the G-action on the product spaces Γ(HN ) ∼= Γ(H)N
with the product measure γHN
∼= γ⊗NH is also ergodic. �
Remark 5.27. Let H be a complex Hilbert space and (π,H) be a unitary representation of G
on H. We write HR for the underlying real Hilbert space and FC(H) for the associated Fock
space over C. Then
Γ
(
HR) ∼= FC
((
HR)
C
) ∼= FC(H⊕H) ∼= FC(H)⊗C FC(H) ∼= B2(FC(H)).
Therefore the requirement that FC(H)G = CΩ is weaker than the ergodicity of the measure γH.
Let H = Hf ⊕ H⊥f denote the decomposition into the closed subspace Hf generated by all
finite-dimensional invariant subspaces and its orthogonal complement H⊥f .7 Then
FC(H) ∼= FC(Hf )⊗FC
(
H⊥f
)
,
6We thank Bachir Bekka for this reference.
7Note that the representation on Hf factors through a representation of a compact group.
Reflection Positive Stochastic Processes Indexed by Lie Groups 43
and Lemma 5.24 implies that the subspace FC(Hf )⊗
⊕
n>0 S
n(H⊥f ) contains no non-zero fixed
vectors. Therefore FC(H)G = FC(Hf )G.
If G is abelian, then Hf is spanned by eigenvectors for certain characters X ⊆ Ĝ, and
the corresponding characters of Sn(Hf ) are the finite products χ1 · · ·χn, χj ∈ X. Therefore
FC(H)G = CΩ is equivalent to the condition that all products χ1 · · ·χn, χj ∈ X, n > 0, are
non-trivial.
Definition 5.28 ([9, Definition 2.14]). A measure preserving action of G on a finite measure
space (X,Σ, µ) is said to be weakly mixing if the representation on the subspace
L2
0(X,Σ, µ)0 := 1⊥ ⊆ L2(X,Σ, µ)
contains no non-zero finite-dimensional invariant subspaces. In view of Theorem 5.26, this is
equivalent to the representation on L2
0(X,Σ, µ) to be weakly mixing in the sense of Defini-
tion 5.25.
The following proposition justifies Definition 5.25(b).
Proposition 5.29. A measure preserving action of G on (X,Σ, µ) is weakly mixing if and only
if the corresponding action on the pair space (X2,Σ⊗ Σ, µ⊗ µ) is ergodic.
Proof. We have
L2(X ×X,µ⊗ µ)G ∼= (L2(X,µ)⊗ L2(X,µ))G ⊆ L2(X,µ)f ⊗ L2(X,µ)f .
If the action is weakly mixing, then L2(X,µ)f = C1 implies that L2(X ×X,µ ⊗ µ)G = C1, so
that the product action is ergodic.
If, conversely, the product action is ergodic and F ⊆ L2
0(X,µ) is a finite-dimensional invariant
subspace, then
F ⊗ F∗ ⊆ L2(X,µ)⊗ L2(X,µ) ∼= L2(X ×X,µ⊗ µ)
leads to a fixed vector in L2(X×X,µ⊗µ), which implies F = {0}. We conclude that the action
on (X,µ) is weakly mixing. �
Example 5.30. If (π,H) is an orthogonal representation for which the corresponding gaussian
action on Γ(H) = (Ha, γH) is ergodic, then it is weakly mixing by Theorem 5.26.
A Continuity of a stochastic process on G
This appendix refers to Example 3.13. Clearly, the most natural continuity requirement from
the perspective of representation theory is that the representation of G in L2(BG,BG, ν) is
continuous. In this subsection we collect some remarks that are useful for the verification of
this continuity.
Lemma A.1. For a semigroup (Ps)s∈S of positivity preserving operators on L∞(X,S, µ), the
strong continuity of the representation of S on L2(X,S, ν) implies continuity in measure
lim
s→s0
ν(|Psf − Ps0f | ≥ ε) = 0 for every ε > 0.
Proof. This follows from ε2ν(|Psf − Ps0f | ≥ ε2) ≤
∫
X |Psf − Ps0f |
2 dν → 0. �
44 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Corollary A.2. For a square integrable stationary R-valued process (Xg)g∈G, the continuity of
the representation of G on L2(BG,BG, ν) implies that
lim
g→1
ν(|Xg −X1| ≥ ε) = 0 for every ε > 0.
Remark A.3. Let G be a group acting in a measure preserving way on the finite measure space
(Q,Σ, µ). It is easy to see that the continuity of the representation on L2(Q,Σ, µ) is equivalent
to the continuity of the orbit maps of the characteristic functions, which in turn is equivalent to
the continuity of the maps
dA : G→ R, dA(g) := µ((gA)∆A) for A ∈ Σ.
Actually this condition is equivalent to the continuity of the G-action on the metric space
(Σ/Jµ, d), where d(A,B) = µ(A∆B) and Jµ denotes the ideal of µ-zero sets.
Next we observe that the set of all bounded functions f ∈ L2(Q,Σ, µ) for which the G-orbit
map in L2(Q,Σ, µ) is continuous is a subalgebra. This follows immediately from the estimate
‖Ug(fh)− fh‖2 ≤ ‖h‖∞‖Ugf − f‖2 + ‖f‖∞‖Ugh− h‖2.
This implies that
Σc := {A ∈ Σ: dA ∈ C(G,R)}
contains ∅ and Q and is stable under complements, finite intersections and finite unions. From
the closedness of the subspace of continuous vectors in L2(Q,Σ, µ) we further derive that Σc is
stable under countable unions, hence a σ-subalgebra. We conclude that it suffices to verify the
continuity of the functions dA for a collection of subsets generating the σ-algebra Σ.
Lemma A.4. For a real-valued stationary process (Xg)g∈G, the condition
lim
g→1
ν(|Xg −X1| ≥ ε) = 0
of continuity in measure implies continuity of the G-representation on L2(Q,Σ, ν).
Proof. In view of the preceding remark, it suffices to show that, for the sets Ag0 := {Xg0 ≥ a},
a ∈ R, the function dA(g) = ν(gAg0∆Ag0) = ν(Agg0∆Ag0) is continuous in 1. Note that
Agg0∆Ag0 ⊆ {|Xgg0 −Xg0 | ≥ δ} ∪ {Xgg0 ∈]a, a+ δ[} ∪ {Xg0 ∈]a, a+ δ[}.
For δ sufficiently small, the last two sets on the right have measure at most ε
3 , so that
ν({|Xg0 −Xgg0 | ≥ δ}) <
ε
2
leads to ν(gAg0∆Ag0) < ε. �
B Markov kernels
In this appendix we discuss briefly some basic properties of Markov kernels that are needed for
this article.
Definition B.1 ([7, Section 36]). (a) Let (Q,Σ) and (Q′,Σ′) be measurable spaces. Then
a function
K : Q× Σ′ → [0,∞]
is called a kernel if
Reflection Positive Stochastic Processes Indexed by Lie Groups 45
(K1) for every A′ ∈ Σ′, the function KA′(ω) := K(ω,A′) is Σ-measurable, and
(K2) for every ω ∈ Q, the function Kω(A′) := K(ω,A′) is a (positive) measure.
A kernel is called a Markov kernel if the measures Kω are probability measures.
(b) A kernel K : Q× Σ′ → [0,∞] associates to a measure µ on (Q,Σ) the measure
(µK)(A′) :=
∫
µ(dω)K(ω,A′)
on (Q′,Σ′). To every measurable function f ′ : Q′ → [0,∞], it associates the function
Kf ′ : Q→ [0,∞], (Kf ′)(ω) :=
∫
Q′
K(ω, dω′)f ′(ω′).
Now the Markov property corresponds to K1 = K for the constant function 1.
(c) If (Qj ,Σj)j=1,2,3 are measurable spaces, then composition of kernels K1 on Q1 × Σ2 and
K2 on Q2 × Σ3 is defined by
(K1K2)(ω1, A3) =
∫
Q2
K1(ω1, dω2)K2(ω2, A3).
In particular, we obtain on a measurable space (Q,Σ) the concept of a semigroup (Ps)s∈S
of (Markov) kernels by the requirement that PsPt = Pst for st ∈ S. Here the classical case is
S = R+.
Remark B.2. (a) ForQ = Q′ and Σ = Σ′, every Markov kernelK defines a positivity preserving
operator on measurable functions by
(Kf)(ω) :=
∫
Q
K(ω, dω′)f(ω′).
The Markov property implies that K1 = 1 and that ‖Kf‖∞ ≤ ‖f‖∞.
(b) For a measure µ on (Q,Σ) we then have∫
Q
f d(µK) =
∫
Q
∫
Q
µ(dω)K(ω, dω′)f(ω′) =
∫
Q
Kfdµ.
Therefore the relation µK = µ is equivalent to the invariance of µ as a functional on non-negative
bounded measurable functions under the operator f 7→ Kf .
Remark B.3. For a Markov semigroup (Pt)t≥0 on (Q,Σ) and a probability measure µ on
(Q,Σ), we obtain for 0 ≤ t1 < · · · < tn and t = (t1, . . . , tn) a probability measure Pµt on Qn [7,
Satz 36.4]:
Pµt (B) =
∫
Qn+1
χB(x1, . . . , xn)µ(dx0)Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn).
The measure Pµt can also be written as Pµt = µPt for the kernel
Pt(x0, B) =
∫
Qn
χB(x1, . . . , xn)Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn)
on Q × Σn. This is a projective family of measures. If (Q,Σ) is a polish space, then this leads
to a stochastic process (Xt)t≥0 with state space (Q,Σ) [7, Corollary 35.4]. The probability
measure µ is the distribution of X0. It is called the initial distribution. According to [7,
Theorem 42.3], the so obtained process has the Markov property.
46 P.E.T. Jorgensen, K.-H. Neeb and G. Ólafsson
Extending stationary Markov processes to the real line
Let (Pt)t≥0 be a Markov semigroup on (Q,Σ) and ν be a measure on (Q,Σ). We obtain for
−sm < −sm−1 < · · · < −s1 < 0 ≤ t1 < · · · < tn
a measure P νs,t on Qm+n by
P νs,t(B) =
∫
Qn+m+1
χB(ym, . . . , y1, x1, . . . , xn)
× Psm−sm−1(ym−1, dym) · · ·Ps2−s1(y1, dy2)Ps1(x0, dy1)ν(dx0)
× Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn).
This means that P νs,t = νPs,t for the kernel
Ps,t(x0, B) =
∫
Qn+m
χB(ym, . . . , y1, x1, . . . , xn)
× Psm−sm−1(ym−1, dym) · · ·Ps2−s1(y1, dy2)Ps1(x0, dy1)
Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn).
This is a projective family of measures. If (Q,Σ) is a polish space, this leads to a stochastic
process (Xt)t∈R with state space (Q,Σ) [7, Corollary 35.4]. If ν is a probability measure, then
the measure ν is called the initial distribution of the process. It coincides with the distribution
of X0. Suppose, in addition, that∫
Q2
g(ω)ν(dω)Pt(ω, dω
′)f(ω′) =
∫
Q
(Ptf)g ν(dx0) =
∫
Q
f(Ptg)ν(dx0)
=
∫
Q2
g(ω)ν(dω′)Pt(ω
′, dω)f(ω′) for 0 ≤ f, g.
Then
P νs,t(B) =
∫
Qn+m+1
χB(ym, . . . , y1, x1, . . . , xn)
× Psm−sm−1(ym−1, dym) · · ·Ps2−s1(y1, dy2)Ps1(x0, dy1)ν(dx0)
× Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn)
=
∫
Qn+m+1
χB(ym, . . . , y1, x1, . . . , xn)
× Psm−sm−1(ym−1, dym) · · ·Ps2−s1(y1, dy2)ν(dy1)Ps1(y1, dx0)
× Pt1(x0, dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn)
=
∫
Qn+m
χB(ym, . . . , y1, x1, . . . , xn)
× Psm−sm−1(ym−1, dym) · · ·Ps2−s1(y1, dy2)ν(dy1)Ps1+t1(y1, dx1)
× Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn)
=
∫
Qn+m
χB(ym, . . . , y1, x1, . . . , xn)
× Psm−sm−1(ym−1, dym) · · · ν(dy2)Ps2−s1(y2, dy1)Ps1+t1(y1, dx1)
× Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn) = · · ·
=
∫
Qn+m
χB(ym, . . . , y1, x1, . . . , xn)
Reflection Positive Stochastic Processes Indexed by Lie Groups 47
× ν(dym)Psm−sm−1(ym, dym−1) · · ·Ps2−s1(y2, dy1)Ps1+t1(y1, dx1)
× Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn).
If ν is a probability measure, we thus obtain a stationary process with values in Q. For t =
(t1, . . . , tn) and t1 < . . . < tn in R, we then have for the distribution of this process
P νt (B) =
∫
Qn
χB(x1, . . . , xn) ν(dx1)Pt2−t1(x1, dx2) · · ·Ptn−tn−1(xn−1, dxn).
This formula immediately implies that the translation invariance of the measure P ν on QR.
Acknowledgements
The research of P. Jorgensen was partially supported by the Binational Science Foundation
Grant number 2010117. The research of K.-H. Neeb was supported by DFG-grant NE 413/7-2,
Schwerpunktprogramm “Darstellungstheorie”. The research of G. Ólafsson was supported by
NSF grant DMS-1101337. The authors wish to thank the Mathematisches Forschungsinsti-
tut Oberwolfach for hosting a Workshop on “Reflection Positivity in Representation Theory,
Stochastics and Physics” November, 30 – December 6, 2014. The present research was started
at the workshop, and it has benefitted from our discussions with the participants there.
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1 Introduction
2 Reflection positive representations
3 Reflection positive Lie group actions on measure spaces
3.1 (G,S,)-measure spaces
3.2 Stochastic processes indexed by Lie groups
3.3 Associated positive semigroup structures and reconstruction
3.4 Klein's characterization of the Markov case
3.5 Total subsemigroups of Lie groups
3.6 The discrete case
4 Measures on path spaces for topological groups
4.1 One-parameter convolution semigroups of measures on polish groups
4.2 Standard path space structures for locally compact groups
5 Gaussian (G,S,)-probability spaces
5.1 Second quantization and gaussian processes
5.2 Application to reflection positive representations
5.3 Equivalence of gaussians measures for reproducing kernel Hilbert spaces
5.4 Gaussian measures on distributions on Lie groups
5.5 Ergodicity of H for G
A Continuity of a stochastic process on G
B Markov kernels
References
References
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