Non-Commutative Vector Bundles for Non-Unital Algebras
We revisit the characterisation of modules over non-unital C∗-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module co...
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irk-123456789-1486412019-02-19T01:31:49Z Non-Commutative Vector Bundles for Non-Unital Algebras Rennie, A. Sims, A. We revisit the characterisation of modules over non-unital C∗-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical index and finite Watatani index. 2017 Article Non-Commutative Vector Bundles for Non-Unital Algebras / A. Rennie, A. Sims // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 57R22; 46L85 DOI:10.3842/SIGMA.2017.041 http://dspace.nbuv.gov.ua/handle/123456789/148641 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We revisit the characterisation of modules over non-unital C∗-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical index and finite Watatani index. |
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Article |
| author |
Rennie, A. Sims, A. |
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Rennie, A. Sims, A. Non-Commutative Vector Bundles for Non-Unital Algebras Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Rennie, A. Sims, A. |
| author_sort |
Rennie, A. |
| title |
Non-Commutative Vector Bundles for Non-Unital Algebras |
| title_short |
Non-Commutative Vector Bundles for Non-Unital Algebras |
| title_full |
Non-Commutative Vector Bundles for Non-Unital Algebras |
| title_fullStr |
Non-Commutative Vector Bundles for Non-Unital Algebras |
| title_full_unstemmed |
Non-Commutative Vector Bundles for Non-Unital Algebras |
| title_sort |
non-commutative vector bundles for non-unital algebras |
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Інститут математики НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/148641 |
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Non-Commutative Vector Bundles for Non-Unital Algebras / A. Rennie, A. Sims // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT renniea noncommutativevectorbundlesfornonunitalalgebras AT simsa noncommutativevectorbundlesfornonunitalalgebras |
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2025-07-12T19:51:28Z |
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2025-07-12T19:51:28Z |
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1837472045707821056 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 041, 12 pages
Non-Commutative Vector Bundles
for Non-Unital Algebras
Adam RENNIE and Aidan SIMS
School of Mathematics and Applied Statistics,
University of Wollongong, Northfields Ave 2522, Australia
E-mail: renniea@uow.edu.au, asims@uow.edu.au
Received December 13, 2016, in final form June 12, 2017; Published online June 16, 2017
https://doi.org/10.3842/SIGMA.2017.041
Abstract. We revisit the characterisation of modules over non-unital C∗-algebras analogous
to modules of sections of vector bundles. A fullness condition on the associated multiplier
module characterises a class of modules which closely mirror the commutative case. We also
investigate the multiplier-module construction in the context of bi-Hilbertian bimodules,
particularly those of finite numerical index and finite Watatani index.
Key words: Hilbert module; vector bundle; multiplier module; Watatani index
2010 Mathematics Subject Classification: 57R22; 46L85
1 Introduction
The Serre–Swan theorem says that the Hilbert modules over unital commutative C∗-algebras
that can be realised as the modules of sections of locally trivial vector bundles over compact
spaces are precisely the finite projective modules. By direct analogy, we define non-commutative
vector bundles over unital C∗-algebras to be finite projective modules, this definition being
justified by the connection to the non-commutative definition of K-theory.
This note revisits the question of the correct notion of a non-commutative vector bundle over
a non-unital C∗-algebra. We prove the equivalence of several conditions on a Hilbert module E
over a non-unital C∗-algebra A that, when A is commutative, characterise modules of sections
of vector bundles. This extends previous partial characterisations [9]. In particular, our results
apply to suspensions of finitely generated projective modules over unital C∗-algebras. This is
an important motivation since it allows us to apply techniques like those of [10] to study Cuntz–
Pimsner algebras of suspended C∗-correspondences, and thereby to bootstrap computational
techniques from even K-groups and Kasparov groups to their odd counterparts.
We then investigate the structure of multiplier modules associated to bimodules that are bi-
Hilbertian in the sense of [6]. We prove that, under mild hypotheses, the bi-Hilbertian structure
and finite numerical index pass to the multiplier module. We also establish that, by contrast, the
multiplier module frequently does not have finite Watatani index. Indeed the multiplier module
has finite Watatani index if and only if it is finitely generated and projective as a right-Hilbert
module over the multiplier algebra, which in turn holds if and only if it is full as a left-Hilbert
module over the multiplier algebra.
We start by briefly recalling what is already known. Then we prove our first result, Theo-
rem 3.1, which shows that if E is a Hilbert module over a σ-unital C∗-algebra A, then its
multiplier module is full as a left-Hilbert EndA(E)-module if and only if E is a finitely generated
projective module over a suitable unitisation of A. We discuss some consequences of this result.
In particular, by applying our results to the setting of commutative C∗-algebras, we prove
that every locally trivial vector bundle V over a locally compact space X of finite topological
mailto:renniea@uow.edu.au
mailto:asims@uow.edu.au
https://doi.org/10.3842/SIGMA.2017.041
2 A. Rennie and A. Sims
dimension extends to a locally trivial vector bundle V c over some compactification Xc of X.
The compactification Xc required, and the isomorphism class of the extension V c, depend on
a choice of frame for V , which we illustrate by example.
We then recall the notion of a bi-Hilbertian bimodule [6]. We prove that if E is countably
generated with injective left action and finite Watatani index, then the bi-Hilbertian structure,
and also finite numerical index, pass from E to its multiplier module in a natural way. We
then describe a number of conditions that are equivalent to the multiplier module being finitely
generated and projective, and show how this applies to modules over commutative C∗-algebras.
2 Finite projective modules and non-unital analogues
Throughout the paper, A denotes a σ-unital C∗-algebra. Given a right C∗-A-module E, we
denote the C∗-algebra of adjointable operators on E by EndA(E). For e, f ∈ E, the rank-1
endomorphism g 7→ e · (f | g)A is denoted by Θe,f , and is adjointable with adjoint Θf,e. We
write End0
A(E) for span{Θe,f : e, f ∈ E}, the closed ∗-ideal of compact endomorphisms in
EndA(E). We write `2(A) for the standard C∗-module over A; that is `2(A) is equal to{
ξ : N → A |
∞∑
n=1
ξ∗i ξi converges in A
}
with inner product (ξ | η)A =
∑
i ξ
∗
i ηi. We say that
a (right) inner product module E over a C∗-algebra A is full if the closed span of the inner
products (e | f)A is all of A.
In keeping with the preceding paragraph, throughout the paper, by a vector bundle over
a locally compact Hausdorff space X, we will mean a locally trivial finite-rank complex vector
bundle equipped with a continuous family of inner products. If X is paracompact, then every
finite rank vector bundle over X admits such a family of inner products.
A frame for a right C∗-A-module E is a sequence {ej}j≥1 ⊂ E such that∑
j≥1
Θej ,ej converges strictly to IdE .
If {ej}j≥1 is a frame for E, then E is generated as a right A-module by the ej , so it is countably
generated.
Any frame {ej} for E determines a stabilisation map in the sense of Kasparov: there is an
adjointable map v : E → `2(A) such that
v(e) =
(
(ej | e)A
)
j≥1 for all e ∈ E.
We have v∗v = IdE , and so p := vv∗ is a projection in EndA(`2(A)); specifically, p satisfies
(pξ)i =
∑
j
(ei | ej)Aξj for all ξ ∈ `2(A).
One of the fundamental points of contact between non-commutative geometry and classical
geometry is the celebrated Serre–Swan theorem. Given a vector bundle V → X over a compact
space X, we write Γ(X,V ) for the space of continuous sections of V .
Theorem 2.1 ([11]). Let X be a compact Hausdorff space. A (right) C(X)-module M is finitely
generated and projective if and only if there is a vector bundle V → X such that M ∼= Γ(X,V ).
We can remove the word “projective” from the statement of Theorem 2.1 if we instead consider
full inner-product modules over C(X) – or more generally inner-product modules E such that
(E |E)C(X) is a unital, and hence complemented, ideal of C(X). This is because every full
finitely generated right inner product C(X)-module can be made into a C∗-module M which,
by Kasparov’s stabilisation theorem, is automatically projective.
Non-Commutative Vector Bundles for Non-Unital Algebras 3
When X is non-compact, the natural C0(X)-module arising from a vector bundle V → X is
the module Γ0(X,V ) of continuous sections of V such that x 7→ (ξ(x) | ξ(x)) belongs to C0(X).
Important examples of this are restrictions V → X of vector bundles V c → Xc over some
compactification Xc of X. The following non-unital Serre–Swan theorem characterises the
algebraic structure of such modules.
Theorem 2.2 ([9, Theorem 8]). Let X be a locally compact Hausdorff space and Xc a compacti-
fication of X. Set A = C0(X) and Ab = C(Xc). A right A-module E is of the form pAn for
some projection p ∈Mn(Ab) if and only if there is a (locally trivial) vector bundle V → Xc such
that E ∼= Γ0(X,V |X).
Theorem 2.2 provides a reasonable algebraic characterisation of modules of the form Γ0(X,V )
where the vector bundle V extends to a bundle over some compactification of X. We use this
to motivate our definition of the non-commutative analogue.
Recall that a unitisation of a nonunital C∗-algebra A is an embedding ι : A ↪→ Ab of A as an
essential ideal of a unital C∗-algebra Ab.
Definition 2.3. Let A be a nonunital separable C∗-algebra and ι : A ↪→ Ab a unitisation of A.
An Ab-finite projective A-module is a right A-module that is isomorphic to pAn for some n ∈ N
and some projection p ∈Mn(Ab).
To state our main result, we need to recall how to embed a Hilbert A-module E in the
associated multiplier module, a tool used frequently in the Gabor-analysis literature (see, for
instance, [1, 7]).
Definition 2.4. Given a right C∗-A-module E, recall that the linking algebra L(E) is the
algebra
L(E) = End0
A(E ⊕A) =
{(
T e
f a
)
: T ∈ End0
A(E), a ∈ A, e ∈ E, f ∈ E
}
,
where E is the conjugate module (a left A-module). Let r = IdE ⊕ 0 ∈ Mult(L(E)) and
s = 0⊕ 1Mult(A) ∈ Mult(L(E)) and define
Mult(E) = rMult(L(E))s.
Then Mult(E) is a right C∗-Mult(A)-module with right action implemented by right multipli-
cation in Mult(L(E)), and right inner-product (e | f)Mult(A) = e∗f .
Writing HomA(A,E) for the Banach space of adjointable operators from A to E, we then
have Mult(E) ∼= HomA(A,E): for any m ∈ Mult(E), the map a 7→ ma belongs to HomA(A,E),
and conversely if m ∈ HomA(A,E), then there is a multiplier of L(E) such that
m
(
T e
f a
)
=
(
m ◦ (f | ·)A ma
0 0
)
.
For e, f ∈ Mult(E), let Θe,f be the associated rank-one operator on Mult(E). We show that
E ⊆ Mult(E) is invariant for Θe,f . To see this, fix g ∈ E, use Cohen factorisation to write
g = g′ · a for some g′ ∈ E and a ∈ A. Regarding g′ as an element of Mult(E), we calculate
Θe,f (g) = e · (f | g′ · a) = e · (f | g)a ∈ Mult(E) ·A, (2.1)
which is equal to E by [1, Remark 3.2(a)].
We deduce that Θe,f |E is adjointable with adjoint Θf,e|E . Hence Θe,f |E ∈ EndA(E). Abusing
notation a little, we will just write Θe,f for this operator on E henceforth. The left inner product
4 A. Rennie and A. Sims
End0Mult(A)(Mult(E))(e | f) = Θe,f gives a norm on Mult(E) equivalent to that coming from the right
Mult(A)-valued inner product. This can be seen directly from the linking algebra picture above.
The inclusion End0
Mult(A)(Mult(E)) ↪→ EndA(E) is injective, and so we can regard Mult(E)
as a left EndA(E)-C∗-module. We will denote the left inner product by EndA(E)(· | ·); we then
have EndA(E)(e | f) = Θe,f |E for e, f ∈ Mult(E).
3 Finite projectivity and the multiplier module
Theorem 3.1. Let A be a non-unital σ-unital C∗-algebra and let E be a right C∗-A-module.
The following are equivalent.
1. There is a finite subset F ⊆ Mult(A) such that, putting Ab := C∗(F ∪ A) ⊆ Mult(A), the
module E is Ab-finite projective.
2. The module E is Mult(A)-finite projective.
3. The module Mult(E) is finitely generated and projective as a right Mult(A) module.
4. The module Mult(E) is full as a left C∗-module over EndA(E).
Proof. The universal property of Mult(A) shows that for any unitisation Ab of A as in (1), we
have an inclusion Ab ↪→ Mult(A). Thus pAnb ⊗Ab
Mult(A) ∼= pMult(A)n and so (1) implies (2).
For (2) implies (1), we observe that since p ∈ Mn(Mult(A)), it has finitely many matrix
entries (pi,j)
n
i=1. Let F = {pi,j : i, j ≤ n}∪{1Mult(A)}, and let Ab = C∗(A∪F ) ⊆ Mult(A). Then
Ab is a unitisation of A, p ∈Mn(Ab), and E ∼= pAn as a right A-module as required.
For (2) implies (3), recall that we may identify Mult(E) with HomA(A,E), from which we
deduce that Mult(pAn) ∼= pMult(A)n.
To see that (3) implies (2), fix a finite frame {ξj} for Mult(E) as a right-Hilbert Mult(A)-
module. By Kasparov’s stabilisation theorem applied to this frame, there exist an integer n ≥ 1,
a projection p ∈Mn(Mult(A)), and an isomorphism ρ : Mult(E)→ pMult(A)n of right Mult(A)-
modules. We claim that ρ(E) = pAn.
To see this, first fix e ∈ E. Apply the strong form [8, Proposition 2.31] of Cohen factorisation
to write e = e′ · (e′ | e′) for some e′ ∈ E. Then ρ(e) = pρ(e) = pρ(e′) · (e′ | e′). Since (e′ | e′) ∈ A
and since Mult(E) · A = E, we deduce that ρ(e) ∈ pAn. So ρ(E) ⊆ pAn. For the reverse
inclusion, we fix a ∈ pAn, and use Cohen factorisation in pAn to write a = a′ · a′′ for some
a′ ∈ pAn and a′′ ∈ A. Write a′ = ρ(ξ) for some ξ ∈ Mult(E). Using again that Mult(E) ·A ⊆ E,
we see that a = ρ(ξ) · a′′ = ρ(ξ · a′′) ∈ ρ(E).
To see that (3) implies (4), suppose that Mult(E) is finitely generated and projective as a right
Mult(A)-module. Then EndMult(A)(Mult(E)) = End0
Mult(A)(Mult(E)), and so EndA(E) ⊆
EndMult(A)(Mult(E)) belongs to the range of the outer product. So Mult(E) is full as a left
C∗-module over EndA(E).
Finally, to see that (4) implies (3), note that associativity of multiplication in Mult(L(E))
shows that
ξ · (η | ζ)Mult(A) = EndA(E)(ξ | η) · ζ for all ξ, η, ζ ∈ Mult(E).
So for ξ, η ∈ Mult(E), the action of EndA(E)(ξ | η) is implemented by the generalised compact op-
erator Θξ,η ∈ End0
Mult(A)(Mult(E)). Since Mult(E) is full as a left EndA(E)-module, the identity
operator IdE is in the range of EndA(E)(· | ·), and is therefore a compact operator on Mult(E).
Since IdE ·e = e = IdMult(E) e for all e ∈ Mult(E), we deduce that IdMult(E) is a compact
operator. So [2, Theorem 8.1.27] implies that Mult(E) is finitely generated. �
Non-Commutative Vector Bundles for Non-Unital Algebras 5
Remark 3.2. Suppose that A and B are C∗-algebras and EA and EB are right-Hilbert modules
over A and B respectively that satisfy the equivalent conditions of Theorem 3.1. Then the
external tensor product (E ⊗ F )A⊗B satisfies the same conditions. For if E ∼= pAn for some
projection p ∈Mn(Mult(A)) and F ∼= qBm with q ∈Mm(B), then E ⊗ F ∼= (p⊗ q)(An ⊗Bm).
Remark 3.3. Suppose that E is an A-module satisfying the equivalent conditions of Theo-
rem 3.1, and that ϕ : A→ B is a non-degenerate ∗-homomorphism. Then we can write E = pAn
for some p ∈ Mult(A). Since ϕ is nondegenerate, it extends to a homomorphism ϕ̃ : Mult(A)→
Mult(B), and we have E⊗ϕB ∼= ϕ̃(p)Bn. Hence E⊗ϕB also satisfies the equivalent conditions
in Theorem 3.1.
The finite set F , and hence the unitisation Ab of A, appearing in Theorem 3.1 is not canonical.
The set F depends on the choice of finite right basis {ξj} for Mult(E) to which Kasparov’s
stabilisation theorem is applied in the second paragraph of the proof of Theorem 3.1. It is
less obvious, but also true, that even if two choices of finite frame {ξj}, {ξ′j} yield the same
unitisation Ab of A, the enveloping projective modules pAnb and p′Anb obtained from these frames
may not be isomorphic, as we now demonstrate.
Example 3.4. We let X := C. Let A = C0(X), and let E = C0(X) the trivial module over
C0(X) with the obvious inner product and multiplication action. Fix any countable locally finite
open cover of X, fix a partition of unity (ϕn)n with respect to this cover, and set en =
√
ϕn for
each n. Then {en} is a frame for E.
The module Mult(E) = Mult(A) is a module over Mult(A) = C(βX), where βX denotes the
Stone–Čech compactification of X. Let us consider two choices of finite frame for this module.
The first choice of frame has a single element
x1 = 1Mult(A).
The construction in the proof of Theorem 3.1 applied to this frame yields Ab = C(C ∪ {∞}) ∼=
C(S2), n = 1, p = 1Ab
and hence pAnb = Ab, the trivial module over Ab.
Now identify Mult(E) with
(
1 0
0 0
)(Mult(A)
Mult(A)
)
, and consider the elements yi ∈ Mult(E) given by
continuous extension of the functions
y1 =
(
1√
1+|z|2
0
)
and y2 =
(
z√
1+|z|2
0
)
to βX. Then (y2 | y2)Mult(A) is identically 1 on the boundary βX \ X, and the other inner
products (yi | yj)Mult(A) are identically zero on the boundary. So all the (yi | yj)Mult(A) take
values in C(X ∪ {∞}), and hence the construction in the proof of Theorem 3.1 applied to this
frame again yields Ab = C(C ∪ {∞}) ∼= C(S2). Now n = 2, and calculation shows that
p =
1
1 + |z|2
(
1 z
z |z|2
)
.
Define w ∈M2(Mult(A)) by continuous extension of the function
w(z) =
1√
1+|z|2
0
z√
1+|z|2
0
to βX. We have
ww∗ =
1
1 + |z|2
(
1 z
z |z|2
)
= p and w∗w =
(
1 0
0 0
)
.
6 A. Rennie and A. Sims
So this w defines an isomorphism of modules pA2 ∼= A over A = C0(X), but this does not extend
to an isomorphism of modules over Ab since w is not well-defined on S2. In particular, pA2
b is
the module of sections of the Hopf line bundle over S2 ∼= C∪ {∞}. This is certainly not trivial,
and so not isomorphic to the trivial extension obtained for the first choice of frame above.
One important situation to which Theorem 3.1 applies is suspensions. Given a bimodule E
over an algebra A we can define the suspended module SE over the suspension SA by
SE = C0(R)⊗ E, (f1 ⊗ a1)(g ⊗ e)(f2 ⊗ a2) = f1gf2 ⊗ a1ea2
and with the obvious inner product.
Corollary 3.5. Let A be a unital C∗-algebra and E a finitely generated Hilbert module over A.
Then Mult(SE) is full as a left-Hilbert EndC(βR)⊗A-module, and so SE is C(βR)⊗A projective.
In fact Mult(SE) is A∼ projective where A∼ is the minimal unitisation.
Proof. Given any frame {ξj} for E, one checks that {1⊗ ξj} is a frame for C(βR)⊗E. So the
result follows from Theorem 3.1. �
The fullness hypothesis in Theorem 3.1 is quite restrictive. In particular, the multiplier
module of a Hilbert module EB need not be full as a left EndA(E)-module. For example, if B is
unital then Mult(E) = E [4], and so if End0
A(E) is non-unital then Mult(E) is not full as a left
EndA(E)-module.
Example 3.6. Let H := `2(N) regarded as a K(H)–C-imprimitivity bimodule, and let H denote
the conjugate C–K(H)-equivalence. Let A = K(H) ⊕ C. Let E := H ⊕H and define an A-bi-
module structure on E by
(T1, λ1)
(
ξ1, ξ2
)
(T2, λ2) =
(
T1ξ1λ2, λ1ξ2T2
)
=
(
T1ξ1λ2, T ∗2 ξ2λ1
)
for all Tj ∈ K(H), λj ∈ C, ξj ∈ H. Define a right inner-product on E by(
(ξ1, ξ2) | (η1, η2)
)
K(H)⊕C =
(
〈ξ1, η1〉,Θξ2,η2
)
.
Since E decomposes as a direct sum of imprimitivity bimodules each of which has a unital
algebra acting on one side or the other, the remark on pages 295 and 296 of [4] shows that E is
equal to its multiplier module Mult(E). The linking algebra is(
A H⊕H
H⊕H A
)
with multiplier algebra(
B(H)⊕ C H⊕H
H⊕H B(H)⊕ C
)
.
Hence Mult(E) = E = H⊕H is not full as a left EndA(E)-module.
4 Application to vector bundles
Recall that for us a vector bundle is always a locally trivial, finite-rank, complex vector bundle
equipped with a continuous family of inner products.
Theorem 4.1. Let X be a second-countable locally compact Hausdorff space of finite topological
dimension.
Non-Commutative Vector Bundles for Non-Unital Algebras 7
a) Suppose that V → X is a vector bundle of rank n. Then there exists a vector bundle Ṽ of
rank n over the Stone–Čech compactification βX such that V ∼= Ṽ |X .
b) There is a metrisable compactification Xc and a vector bundle V c → Xc such that V =
V c|X .
Proof. (a) Let E = Γ0(X,V ), and let A = C0(X). By Theorems 2.2 and 3.1, it suffices to
show that Mult(E) is full as a left EndA(E)-module. Using Mult(E) ∼= HomA(A,E), we see
that every continuous bounded section of V defines an element of Mult(E).
Fix a countable open cover U of X by sets on which V is trivial. Since X has finite topological
dimension, say dim(X) = d, we can assume, by refining if necessary, that there is a partition
U =
d⊔
i=0
Ui such that distinct elements U1, U2 of any given Ui are disjoint. Fix a partition of
unity {hU : U ∈ U} and choose linearly independent sections {fU,j : U ∈ U , j = 1, . . . , n} of V |U
such that
n∑
j=1
EndA(E)(fU,j(x) | fU,j(x)) =
n∑
j=1
ΘfU,j(x),fU,j(x) = hU (x)IdV
for all x, U . For each 0 ≤ i ≤ d and 1 ≤ j ≤ n, the pointwise sum Fi,j :=
∑
U∈Ui
fU,j is a bounded
section of V and so belongs to Mult(E). We then have
IdV =
d∑
i=1
n∑
j=1
EndA(E)(Fi,j |Fi,j),
so Mult(E) is full as a left EndA(E)-module. A similar argument shows that Mult(E) is a full
right C(βX)-module.
For (b), note that since X is second-countable, C0(X) is separable. Thus by part (1) of
Theorem 3.1 we see that E is Ab-finite projective for a separable unitization Ab of C0(X).
So Ab ∼= C(Xc) for some second-countable, and hence metrisable, compactification of X. The
module of sections of the restriction of Ṽ to Xc is then a finite projective module over C(Xc). �
Corollary 4.2. Let A be nonunital, separable and commutative, so A ∼= C0(X) and suppose
that X is of finite topological dimension. Let E be a right C∗-A-module. Then the conditions
(1)–(4) of Theorem 3.1 are equivalent to
5. There is a vector bundle V → X such that E ∼= Γ0(X,V ).
Proof. Suppose E satisfies (5). By Theorem 4.1, the vector bundle V extends to a bundle Ṽ on
the Stone–Čech compactification. By Theorem 2.2, E = Γ0(X, Ṽ |X) ∼= p(C(βX)N ) for some N
and some p ∈ MN (C(βX)), and so E satisfies (1). Conversely, if E is Ab-finite projective for
some unitisation Ab, then Theorem 2.2 implies that there is a vector bundle Ṽ over the spectrum
of Ab such that E ∼= Γ0(X, Ṽ |X). This completes the proof. �
Corollary 4.3. Suppose that X is a locally compact Hausdorff space with finite topological
dimension. Then a right C∗-C0(X)-module E is C(Xc) finite projective for some compactifica-
tion Xc of X if and only if E ∼= Γ0(X,V ) for some vector bundle V → X.
5 Multiplier modules of bi-Hilbertian bimodules
In this section, we investigate the structure of the multiplier module of a bi-Hilbertian bimodule
in the sense of Kajiwara–Pinzari–Watatani (see Definition 5.1 below). We show that if E
8 A. Rennie and A. Sims
has finite Watatani index then its multiplier module can always be made into a bi-Hilbertian
bimodule with finite numerical index in its own right.
We then show that the passage of finite Watatani index from E to Mult(E) is much less com-
mon; it is equivalent, for example, to fullness of Mult(E) as a left-Hilbert EndA(E)-module. In
particular, finite Watatani index does not help to weaken the fullness hypotheses of Theorem 3.1.
Definition 5.1 ([6, Definition 2.3]). Let A be a σ-unital C∗-algebra. A bi-Hilbertian A-bimodule
is a countably generated full right Hilbert C∗-A-module with inner product (· | ·)A which is also
a countably generated full left Hilbert C∗-A-module with inner product A(· | ·) such that the left
action of A is adjointable with respect to (· | ·)A and the right action of A is adjointable with
respect to A(· | ·).
Since a bi-Hilbertian bimodule is complete in the norms coming from both the left and the
right inner products, these two norms are equivalent.
Let E be a bi-Hilbertian A-bimodule. We recall from [6, Definition 2.8] the definition of the
right numerical index of E. We say that E has finite right numerical index if there exists λ > 0
such that∥∥∑
i A(fi | fi)
∥∥ ≤ λ∥∥∑i Θfi,fi
∥∥ for all n and all f1, . . . , fn ∈ E.
The right numerical index of E is the infimum of the numbers λ satisfying the above inequality.
Let E be a bi-Hilbertian A-bimodule, countably generated as a right module and with finite
right numerical index. By [6, Corollaries 2.24 and 2.28], the left action of A on E is by compact
endomorphisms with respect to (· | ·)A if and only if, for every frame {ej} for (E, (· | ·)A), the
series∑
j≥1
A(ej | ej)
converges strictly in Mult(A). In this case we denote the strict limit by r-Ind(E). We note that
r-Ind(E) is independent of the frame {ej}, and is called the right Watatani index of A. This
r-Ind(E) is a positive central element of Mult(A), and is invertible if and only if the left action
of A is implemented by an injective homomorphism A→ End0
A(E).
As in [6], we simply say that E has finite right Watatani index if it has finite right numerical
index and the left action is by compacts.
Theorem 5.2. Let A be a nonunital σ-unital C∗-algebra and let E be a bi-Hilbertian A-bimodule,
countably generated as a right module and with injective left action. Suppose that E has finite
right Watatani index. Then A(· | ·) extends to a strictly continuous Mult(A)-valued inner product
on Mult(E), with respect to which Mult(E) is a bi-Hilbertian Mult(A)-bimodule with finite right
numerical index.
Proof. When the bi-Hilbertian A-bimodule E has finite right Watatani index, [6, Corol-
lary 2.11] shows that there is a positive A-bilinear norm continuous map Φ: End0
A(E) → A
such that
A(e | f) = Φ(Θe,f ), e, f ∈ E.
Proposition 2.27 of [6] implies that Φ extends to a bounded strictly continuous positive A-bilinear
map Φ: EndA(E)→ Mult(A) with ‖Φ‖ ≤ ‖ r-Ind(E)‖.
Now let Mult(E) be the multiplier module of E. The inclusion A ↪→ End0
A(E) ⊆ L(E)
implementing the left action extends to a homomorphism Mult(A)→ EndA(E) ⊆ Mult(L(E)),
giving a left action of Mult(A) on Mult(E) by (· | ·)Mult(A)-adjointable operators.
Non-Commutative Vector Bundles for Non-Unital Algebras 9
We saw at (2.1) that for e, f ∈ Mult(E) the operator Θe,f ∈ End0
Mult(A)(Mult(E)) restricts
to an adjointable operator on E, so we can define Mult(A)(· | ·) : Mult(E)×Mult(E)→ Mult(A)
by
Mult(A)(e | f) := Φ(Θe,f ).
By definition of Φ, this strictly continuous form extends A(· | ·).
We claim that this is a left-Mult(A)-linear Mult(A)-valued inner-product on Mult(E). To see
this, observe that sesquilinearity of (e, f) 7→ Θe,f , linearity of restriction and linearity of Φ show
that Mult(A)(· | ·) is sesquilinear. For a ∈ Mult(A) and e, f ∈ Mult(E), we use the A-bilinearity
of Φ to see that
a · Mult(A)(e | f) = aΦ(Θe,f |E) = Φ(aΘe,f |E) = Φ(Θa·e,f |E) = Mult(A)(a · e | f),
so Mult(A)(· | ·) is left-A-linear. It is positive because Φ is.
By [6, Proposition 2.27(1)], the maximal λ′ > 0 such that λ′‖(e | e)A‖ ≤ ‖A(e | e)‖ for all
e ∈ E satisfies
λ′T ≤ Φ(T ) ≤ ‖T‖ r-Ind(E) for all 0 ≤ T ∈ EndA(E).
Hence Mult(A)(· | ·) is positive definite. The same inequality combined with the norm equality
‖Θe,e‖ = ‖(e | e)Mult(A)‖ gives
λ′‖(e | e)Mult(A)‖ ≤ ‖Mult(A)(e | e)‖ ≤ ‖ r-Ind(E)‖ ‖(e | e)Mult(A)‖,
and so the norms on Mult(E) coming from the left and right inner products are equivalent.
To see that Mult(E) has finite right numerical index, let g1, . . . , gm ∈ Mult(E) be any finite
set. Since Φ is bounded we have∥∥∑
iMult(A)(gi | gi)
∥∥ =
∥∥∑
i Φ(Θgi,gi)
∥∥ ≤ ‖Φ‖ ∥∥∑i Θgi,gi
∥∥.
In particular, given a frame {gj : j ∈ N} for Mult(E) and any finite subset I of N, we have∥∥∑
i∈I Mult(A)(gi | gi)
∥∥ ≤ ‖Φ‖.
So Mult(E) has finite right numerical index. �
Our next theorem shows that while finite right numerical index passes easily from E to
Mult(E) as above, finite right Watatani index is another question.
Theorem 5.3. Let A be a σ-unital non-unital C∗-algebra and let E be a bi-Hilbertian A-bi-
module, countably generated as a right module and with injective left action. Suppose that E has
finite right Watatani index. Then the following are equivalent:
1) Mult(E) is full as a left Mult(A)-module;
2) IdMult(E) is a compact endomorphism of Mult(E)Mult(A);
3) the left action of Mult(A) on Mult(E)Mult(A) is by compact endomorphisms;
4) r-Ind(Mult(E)) ∈ Mult(A) (that is, Mult(E) has finite right Watatani index); and
5) E is Mult(A)-finite projective as in Definition 2.3.
Proof. The left action of Mult(A) is injective. Hence [6, Corollaries 2.20 and 2.28] gives
(1)⇔ (4). We have (2)⇔ (3) because End0
Mult(A)(Mult(E)) is an ideal of EndMult(A)(Mult(E))
and because the identity of Mult(A) acts as IdMult(E). Similarly, (5)⇒ (3) since in this case
all the endomorphisms are compact. The implication (1)⇒ (5) follows from the implication
(4)⇒ (2) in Theorem 3.1. So it suffices to prove that (3)⇔ (4), which follows from [6, Theo-
rem 2.22]. �
10 A. Rennie and A. Sims
Remark 5.4. If the equivalent conditions (1)–(5) of Theorem 5.3 hold, then Theorem 3.1 shows
that in fact E is Ab-finite projective for some subalgebra of Mult(A) generated by A and just
finitely many additional elements.
Remark 5.5. By Theorem 5.2, if the equivalent conditions in Theorem 5.3 hold, then the
left inner product on E extends by strict continuity to a Mult(A)-valued left inner product on
Mult(E) under which Mult(E) itself becomes a bi-Hilbertian bimodule with invertible finite
right Watatani index.
Example 5.6. We describe a concrete example where Theorem 5.2 applies but the equivalent
conditions in Theorem 5.3 do not hold. Consider the non-unital C∗-algebra A := K, the compact
operators on `2(N). Let E be the external tensor product E = `2 ⊗ K, which is a Morita
equivalence from K ∼= K ⊗ K to C ⊗ K ∼= K, and hence an A–A-imprimitivity bimodule. So E
has finite right Watatani index r-Ind(E) = 1M(K).
It is routine to check that Mult(E) ∼= `2 ⊗ B(`2). Since the left action of Mult(A) = B(`2)
on this module is not by compact operators, Theorem 5.3(2) fails, and therefore so do the other
conditions. In particular, while Mult(E) has finite numerical index by Theorem 5.2, it does not
have finite right Watatani index.
Finite right Watatani index can nevertheless be a useful tool for deciding when we obtain
finite projective modules over unitisations. For a commutative algebra A, we can equip a right
inner product module E over A with a left action and inner product via the formulae
a · e := ea, and A(e1 | e2) := (e2 | e1)A, for all e, e1, e2 ∈ E, a ∈ A. (5.1)
So we are in the bi-Hilbertian setting. For each character φ ∈ Â, the (completion of the) quotient
Hφ := E/(E · kerφ) is a Hilbert space with inner product 〈e1, e2〉 = φ((e1 | e2)A).
Corollary 5.7. Let X be a second-countable locally compact Hausdorff space of finite topological
dimension, and let A = C0(X). Let E be a full right Hilbert A-module, regarded as a bi-Hilbertian
bimodule as in (5.1). The following are equivalent:
1) The map φ 7→ dimHφ is a continuous bounded function from X to [0,∞);
2) E is isomorphic to Γ0(X,V ) for some (finite rank) vector bundle V → X;
3) E has finite right Watatani index.
Proof. We first prove that (1) implies (2). For e ∈ E, define a section Se : X →
⋃
φ∈X Hφ =
E/(E · kerφ) by Se(φ) = e + E · kerφ. This set of sections is a vector space and is fibrewise
dense (in fact, fibrewise surjective), and so by [5, Theorem II.13.18] there is a unique topology
on V :=
⋃
φHφ under which these sections are continuous.
For a ∈ C0(X) and e ∈ E, we see that Se·a(φ) = a(φ)Se(φ) for all φ ∈ Â. So [5, Corol-
lary II.14.7] shows that the Se are uniformly-on-compacta dense in Γ0(X,V ). A simple argu-
ment using completeness of E then shows that e 7→ Se is a surjection of E onto Γ0(X,V ). By
definition, this surjection carries the right action and inner product on E to those on Γ0(X,V ).
So e 7→ Se is an isomorphism E ∼= Γ0(X,V ).
We must check that V is locally trivial. Remark II.13.9 of [5] states that finite-rank continuous
bundles of Banach spaces of constant dimension are locally trivial; we provide a proof in our
setting for completeness (see also [3, Remarque, p. 231]).
Fix φ ∈ X, and choose an orthonormal basis {e1, . . . , en} for Hφ. Fix elements ξi ∈ E such
that ξi + E · kerφ = ei. By scaling by an appropriate element of C0(X) we can assume that
there is a neighbourhood U of φ such that for ψ ∈ U , the element eψi := ξi + E · kerψ satisfies
‖eψi ‖ = 1. Since ψ 7→ dimHψ is continuous, and since (ξi | ξj) is continuous for all i, j, by
Non-Commutative Vector Bundles for Non-Unital Algebras 11
shrinking U if necessary we can assume that dimHψ = n for all ψ ∈ U and that |〈eψi , e
ψ
j 〉| <
1
2n2
for i 6= j and ψ ∈ U . Now if ψ ∈ U and
∑
i αie
ψ
i = 0, then
0 =
∣∣∣〈∑
i
αie
ψ
i ,
∑
j
αje
ψ
j
〉∣∣∣ ≥∑
i
|αi|2‖eψi ‖
2 − (n2 − n) maxi |αi|2
2n2
≥ max
i
|αi|2/2.
So V admits a continuous choice of basis on U , so is locally trivial.
To see that (2) implies (3), suppose that E has the form Γ0(X,V ). Choose a locally finite
cover of X by sets Ui on which on which V is trivial, say V |Ui
∼= Ui × Cni . Since V is finite
rank, supi ni < ∞. Let e1, . . . , ej denote the bounded sections of V |Ui given by the standard
basis of Cni .
Choose a partition of unity (hi) subordinate to the Ui, and put fi,j =
√
hiej for 1 ≤ j ≤ ni
and i ∈ N. Then just as in Theorem 4.1 we see that∑
i,j
Θfi,j ,fi,j =
∑
i
hiIdV |Ui
= IdV .
Using the module structure defined in (5.1), we see that the left inner product of two sections
e =
∑
j cjej , f =
∑
k dkek supported in Ui is given by
A(e | f) =
∑
j,k
cjdk(ek | ej)A =
∑
j
cjdj =
∑
j
(ej | e)A(f | ej)A =
∑
j
(ej |Θe,fej)A
= Trace(Θe,f ).
Hence for any finite set {ek : k ≤ K} ⊆ E, we have∥∥∥∑
k
A(ek | ek)
∥∥∥ ≤ (sup
i
ni)
∥∥∥∑
k
Θek,ek
∥∥∥,
and so E has finite numerical index.
This also shows that
∑
j A(fi,j | fi,j) = hini on the open set Ui, and so
(∑
i,j A(fi,j | fi,j)
)
a
converges to nia for a ∈ Cc(Ui). So
∑
i,j A(fi,j | fi,j) converges strictly. That is, E has finite
right Watatani index.
Finally, for (3) implies (1), suppose that E has finite right Watatani index. By definition
of the bi-Hilbertian structure (5.1), the right Watatani index of E is the function φ 7→ dimHφ.
By definition of finite Watatani index, we deduce that (φ 7→ dimHφ) ∈ Mult(C0(X)) ∼= Cb(X),
giving (1). �
Acknowledgements
This research was supported by Australian Research Council grant DP150101595. It was moti-
vated by questions arising in projects with our collaborators Francesca Arici, Magnus Goffeng,
Bram Mesland and Dave Robertson, and we thank them for all that we have learned from
them. We are very grateful to the anonymous referee who read the manuscript very closely and
made numerous very helpful suggestions that have significantly strengthened our results and
streamlined our proofs. Thanks, whoever you are.
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https://doi.org/10.2307/1993627
1 Introduction
2 Finite projective modules and non-unital analogues
3 Finite projectivity and the multiplier module
4 Application to vector bundles
5 Multiplier modules of bi-Hilbertian bimodules
References
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