On Isomorphism Between Certain Group Algebras on the Heisenberg Group
Let IHn denote the (2n + 1)-dimensional Heisenberg group and let K be a compact subgroup of Aut(IHn); the group of automorphisms of IHn. We prove that the algebra of radial functions on IHn and the algebra of spherical functions arising from the Gelfand pairs of the form (K, IHn) are algebraically i...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1067432016-10-05T03:02:04Z On Isomorphism Between Certain Group Algebras on the Heisenberg Group Egwe, M.E. Bassey, U.N. Let IHn denote the (2n + 1)-dimensional Heisenberg group and let K be a compact subgroup of Aut(IHn); the group of automorphisms of IHn. We prove that the algebra of radial functions on IHn and the algebra of spherical functions arising from the Gelfand pairs of the form (K, IHn) are algebraically isomorphic. Пусть IHn обозначает (2n+1)-мерную группу Гейзенберга, а K - компактную подгруппу Aut(IHn), группу автоморфизмов IHn. Доказано, что алгебра радиальных функций на IHn и алгебра сферических функций, возникающих из пар Гельфанда вида (K, IHn), являются алгебраически изоморфными. 2013 Article On Isomorphism Between Certain Group Algebras on the Heisenberg Group / M.E. Egwe, U.N. Bassey // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 150-164. — Бібліогр.: 20 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106743 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Let IHn denote the (2n + 1)-dimensional Heisenberg group and let K be a compact subgroup of Aut(IHn); the group of automorphisms of IHn. We prove that the algebra of radial functions on IHn and the algebra of spherical functions arising from the Gelfand pairs of the form (K, IHn) are algebraically isomorphic. |
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Egwe, M.E. Bassey, U.N. |
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Egwe, M.E. Bassey, U.N. On Isomorphism Between Certain Group Algebras on the Heisenberg Group Журнал математической физики, анализа, геометрии |
| author_facet |
Egwe, M.E. Bassey, U.N. |
| author_sort |
Egwe, M.E. |
| title |
On Isomorphism Between Certain Group Algebras on the Heisenberg Group |
| title_short |
On Isomorphism Between Certain Group Algebras on the Heisenberg Group |
| title_full |
On Isomorphism Between Certain Group Algebras on the Heisenberg Group |
| title_fullStr |
On Isomorphism Between Certain Group Algebras on the Heisenberg Group |
| title_full_unstemmed |
On Isomorphism Between Certain Group Algebras on the Heisenberg Group |
| title_sort |
on isomorphism between certain group algebras on the heisenberg group |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/106743 |
| citation_txt |
On Isomorphism Between Certain Group Algebras on the Heisenberg Group / M.E. Egwe, U.N. Bassey // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 150-164. — Бібліогр.: 20 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT egweme onisomorphismbetweencertaingroupalgebrasontheheisenberggroup AT basseyun onisomorphismbetweencertaingroupalgebrasontheheisenberggroup |
| first_indexed |
2025-07-07T18:56:01Z |
| last_indexed |
2025-07-07T18:56:01Z |
| _version_ |
1837015564045778944 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 2, pp. 150–164
On Isomorphism Between Certain Group Algebras
on the Heisenberg Group
M.E. Egwe and U.N. Bassey
Department of Mathematics, University of Ibadan
Ibadan, Nigeria
E-mail: me−egwe@yahoo.co.uk
unbassey@yahoo.com
Received July 6, 2009, revised March 27, 2012
Let IHn denote the (2n + 1)-dimensional Heisenberg group and let K
be a compact subgroup of Aut(IHn), the group of automorphisms of IHn.
We prove that the algebra of radial functions on IHn and the algebra of
spherical functions arising from the Gelfand pairs of the form (K, IHn) are
algebraically isomorphic.
Key words: Heisenberg group, spherical functions, radial functions, Heat
kernel, algebra isomorphism.
Mathematics Subject Classification 2010: 43A80, 22E45, 33E99, 33C65.
1. Introduction
In [1], Krotz et al. studied the heat kernel transform for the Heisenberg group
while Sikora and Zienkiewicz [2] described the analytic continuation of the heat
kernel on the Heisenberg group. Earlier, Cowling et al. [3] derived a formula for
the heat semigroup generated by a distinguished Laplacian on a large class of
Iwasawa AN groups and proved that the maximal function constructed from the
semigroup is of weak type (1, 1). Thangavelu [4] studied the spherical mean value
operators Lr on the reduced Heisenberg group IHn/Γ, where Γ is the subgroup
{(0, 2πk) : k ∈ ZZ} of IHn, and showed that all the eigenvalues of the operator Lr
defined by Lrf = αf are of the form ψπ(r) =
∫
G φπ(x)dνr.
In this paper, we show that the algebra of spherical functions generated by
the Gelfand space ∆(K, IHn), the space of bounded K-spherical functions on IHn
modulo its center, equipped with compact-open topology associated to the Lapla-
cian is algebraically isomorphic with the algebra of integrable radial functions on
IHn. This implies that these two algebras can be compared as sets considering
their closed ideals as can be seen in [5]. Here (K, IHn) is a Gelfand pair with
K ⊆ U(n), the group of Aut(IHn).
c© M.E. Egwe and U.N. Bassey, 2013
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
1.1. The Heisenberg group. The (2n + 1)-dimensional Heisenberg group,
IHn, is a noncommutative nilpotent Lie group whose underlying manifold is ICn×
IR with coordinates (z, t) = (z1, z2, . . . , zn, t) and group law given by
(z, t)(z
′
, t
′
) = (z + z
′
, t + t
′
+ 2Imz.z
′
),
where
z.z
′
=
n∑
j=1
zj z̄j , z ∈ ICn, t ∈ IR.
Setting zj = xj+iyj , then (x1, . . . , xn, y1, . . . , yn, t) forms a real coordinate system
for IHn. In this coordinate system, we define the following vector fields:
Xj =
∂
∂xj
+ 2yj
∂
∂t
, Yj =
∂
∂yj
− 2xj
∂
∂t
, T =
∂
∂t
.
It is clear from [6] that {X1, . . . , Xn, Y1, . . . , Yn, T} is a basis for the left invariant
vector fields on IHn and the following commutation relations hold:
[Yj , Xk] = 4δjkT, [Yj , Yk] = [Xj , T ] = [Xj , Xk] = 0.
Similarly, we obtain the complex vector fields by setting
Zj =
1
2
(Xj − iYj) =
∂
∂zj
+ iz̄
∂
∂t
Z̄j =
1
2
(Xj + iYj) =
∂
∂z̄j
− iz
∂
∂t
and we have the commutation relations
[Zj , Z̄k] = −2δjkT, [Zj , Zk] = [Z̄j , Z̄k] = [Zj , T ] = [Z̄, T ] = 0.
The Haar measure on IHn is the Lebesgue measure
dzdz̄dt
on ICn × IR [7]. In particular, for n = 1, we obtain the 3-dimensional Heisenberg
group IH1
∼= IR3 (since ICn ∼= IR2n).
Let us briefly recall the definition and properties of spherical functions which
we shall need in the sequel.
1.2. Basic Definitions. Let G be a semisimple noncompact connected
Lie group with finite center, and K be a maximal compact subgroup. Let
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 151
M.E. Egwe and U.N. Bassey
Cc(K \G/K) denote the space of continuous functions with compact support
on G which satisfy f(k1gk2) = f(g) for all k1, k2 in K. Such functions are called
spherical or K-bi-invariant. Then, Cc(K \ G/K) forms a commutative Banach
algebra under convolution [8]. An elementary spherical function φ is defined to
be a K-bi-invariant continuous function which satisfies φ(e) = 1 and such that
f → f ∗ φ(e) defines an algebra homomorphism of Cc(K \G/K).
The elementary spherical functions are characterized by the following prop-
erties (see [9]):
(i) They are eigenfunctions of the convolution operator
f ∗ φ = φ̂(f)φ,
where
φ̂(f) =
∫
G
f(x−1)φ(x)dx.
(ii) They are eigenfunctions for a large class of left invariant differential opera-
tors on G.
(iii) They satisfy ∫
K
φ(xky)dk = φ(x)φ(y).
Now, on the Heisenberg group, we consider K, a compact group of subgroup
of automorphisms of IHn such that the convolution algebra L1
K of K−invariant
functions is commutative. A bounded continuous K−invariant function ϕ such
that f → ∫
fϕ is an algebra homomorphism on L1
K is called a K−spherical
function. (For a complete characterization of the K−spherical functions and
their properties (for various different K, see [10, 11].) In fact, when K = U(n),
the K−spherical functions include elementary spherical functions.
A function f : IRn −→ IR is said to be radial if there is a function φ defined
on [0,∞) such that f(x) = φ(|x|) for almost every x ∈ IRn.
Simple and classical examples of radial functions and their properties can be
seen in, for example, [12, p. 464],[13, p. 266], [14, p. 134] and [15, p. 366].
Let ρ be a transformation on IRn and x ∈ IRn. Then ρ is said to be orthogonal
if it is a linear operator on IRn that preserves the inner product 〈ρx, ρy〉 = 〈x, y〉
for all x, y ∈ IRn. If detρ = 1, ρ is called a rotation. Hence,
(1) We thus have that from the definition above a function f , defined on IRn,
is radial if and only if f(ρx) = f(x) for all orthogonal transformations ρ
of IRn.
152 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
(2) Also, f is radial if and only if f(ρx) = f(x) for all rotations ρ and all
x ∈ IRn when n > 1.
(3) The basic property of Fourier with respect to orthogonal transformations
is that the Fourier transformation F commutes with orthogonal transfor-
mations, i.e., if ρ is an orthogonal transformation. Let Rρ be the mapping
taking f on IRn into a function g whose values are g(x) = (Rρf)(x) = f(ρx)
for x ∈ IRn, then whenever f ∈ L1(IRn),
ĝ(t) = (Fg)(t) = (FRρf)(t) = (RρFf)(t) = (Ff)(ρt) = f̂(ρt), (1.1)
i.e., the operators F and Rρ commute: FRρ = RρF [21, p. 135].
To see this, we notice that the adjoint of ρ is also its inverse and the Jacobian in
the change of variable ω = ρx is one. Thus we have
ĝ(t) =
∫
IRn
e−2πit.xf(ρx)dx =
∫
IRn
e−2πit.ρ−1ωf(ω)dω
=
∫
IRn
e−2πiρtωf(ω)dω = f̂(ρt). (1.2)
Now, since whenever |x1| = |x2| for two points of IRn, there is an orthogonal
transformation ρ such that ρx1 = x2, we obtain the above mentioned property of
the Fourier transform and thus we have that if f is a radial function in L1(IRn),
then f̂ is also radial [16].
2. Radial Functions On IHn
Let θ = (θ1, . . . , θn) ∈ IRn. We define an automorphism αθ of IHn by
αθ : (z, t) 7→ (eiθz, t) : IHn → Aut(IHn),
where
eiθz = (eiθ1z1, . . . , e
iθnzn).
We then have
Uλ
(eiθz,t)
= A−1
θ Uλ
(z,t)Aθ for λ > 0,
Uλ
(eiθz,t)
= AθU
λ
(z,t)A
−1
θ for λ < 0
(2.1)
where AθF (z) = F (eiθz) and Uλ
(z,t) denotes the irreducible unitary representation
of IHn. Also, for λ = 0, we obtain
χω(eiθz, t) = χe−iθω (z, t),
where χω(z, t) = eiRe〈z,w〉 is the 1-dimensional representation of IHn.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 153
M.E. Egwe and U.N. Bassey
Definition 2.1. A function f, defined on IHn, is said to be radial if
f(z, t) = f(eiθz, t) for all θ. (2.2)
Thus, by 2.0(1), if f is radial, then the operators Uλ
f , λ 6= 0, and Aθ commute
and, since
Aθφ
λ
n = ei〈θ,n〉φλ
n, (2.3)
we have
Uλ
f φλ
n = f̂(λ, n)φλ
n, where f̂(λ, n) ∈ IC. (2.4)
Also, for λ = 0, we write
f̂(0, ρ) =
∫
IHn
f(z, t)χω(z, t)dzdt, (2.5)
where ρ = (|ω1|, · · · , |ωn|).
In what follows, let A denote the space of radial functions in L1(IHn). Now,
since α
θ
are automorphisms of IHn, A is a closed *-subalgebra of L1(IHn). And it
follows from (2.4) that the algebra A is commutative. Since L1(IHn) is symmetric
[17], the *-subalgebra A is also symmetric. The following results are well known.
Proposition 2.2. [18] All non-zero multiplicative functionals on A are either
of the form
(a) f −→ f̂(λ, n) (as in (2.4))
or of the form
(b) f −→ f̂(0, ρ) (as in (2.5)).
P r o o f. Let ψ be a non-zero multiplicative linear functional onA. Since A is
a symmetric *-subalgebra of L2(IHn), there exists an irreducible *-representation
π of L1(IHn) and a unit vector ξ in the Hilbert space Hπ such that πfξ = ψ(f)ξ
for f in A. If Hπ is one-dimensional, then ψ has the form (b). Otherwise, π = Uλ
for some λ 6= 0 and Hπ = Hλ. Since {Uλ
f : f ∈ A} is a *-algebra of operators
which are diagonal on the basis φλ
n, we have ξ = φλ
n for some n and (a) follows.
Proposition 2.3. [18] If f ∈ A, then
f̂(λ, n) =
∫
IHn
f(z, t)e−iλte−|λ||z|
2
r∏
j=1
Lnj (2|λ||zj |2)dzdt, (2.6)
154 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
where Lk is the Laguerre polynomial of degree k, that is,
Lk(x) =
k∑
j=0
(
k
j
)
(−x)j
j!
.
For r > 0, recall that a dilation of IHn is defined by
δr(z, t) = (r−1/2z, r−1t).
δr is an automorphism of IHn and so δr(f)(z, t) = r−n−1f(δr(z, t)) defines an
automorphism of L1(IHn) which preserves A. For a functional ψ on A, let
〈f, δ∗rψ〉 = 〈δrf, ψ〉. δ∗r maps the Gelfand space M(A) of non-zero multiplicative
functionals on A homeomorphically onto itself. On the other hand, if f ∈ L1(IHn)
and
∫
IHn
f(z, t)dzdt = 1, {δrf} is an approximate identity in L1(IHn) as r −→ 0.
Proposition 2.4. A is a (commutative) regular algebra and the set of func-
tions f in A whose Gelfand transform f̂ has support in M(A) is dense in A.
We give some examples of radial functions on IHn.
E x a m p l e 2.5. Let
Dn = {(z, z0) ∈ ICn × IC : Imz0 > |z|2}
on which the Heisenberg group IHn acts by translations [6]
(ω, u)(z, z0) −→ (ω, u) ·(z, z0) = (ω+z, z0+u+i|ω|2+2i〈z, ω〉 : IHn×Dn −→ Dn.
Introducing new coordinates t, ε, z
z0 = t + i(ε + |z|2),
z = z,
Dn
∼= IHn × IR+ and the level surfaces for the variable ε are the orbits of IHn in
Dn. Also, IHn is identified with the boundary ∂Dn of Dn.
Let ∆ be the Laplace–Beltrami operator for the Bargman metric on Dn. The
bounded harmonic functions u on Dn, i.e., ∆u = 0, have boundary values a.e. on
∂Dn, i.e.,
lim
ε→0
u(z, t, ε) = ϕ(z, t) a.e.,
where ϕ ∈ L∞(IHn). Moreover,
u(z, t, ε) = (ϕ ∗ Pε)(z, t),
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 155
M.E. Egwe and U.N. Bassey
where
Pε(z, t) = cnεn+1((|z|2 + ε)2 + t2)−n−1,
Cn =
2r−1n!
πn+1
and the convolution is on IHn.
We notice that Pε ∈ L1(IHn) and is radial. Pε can be expressed as
Pε = c−1
n εn+1|Sε|2,
where
Sε(z, t) = cn(ε + |z|2 − it)−n−1
is the Szego Kernel for Dn, which determines the orthogonal projection of L2(IHn)
on the Hardy space H2(Dn) and precisely the spherical harmonics earlier ob-
tained. It has been shown in [18] that: For every ε > 0. P̂ε does not varnish at
any point in the Gelfand space M(Ar).
We give next the group Fourier transform of radial functions on the Heisenberg
group. Recall that the group Fourier transform of an integrable function g on
IHn is, for each λ 6= 0, an operator-valued function on the Hilbert space L2(IRn)
given by
ĝ(λ)ϕ(ξ) = Wλ(gλ)ϕ(ξ),
where
Wλ(fλ)ϕ(ξ) =
∫
ICn
gλ(z)πλ(z)ϕ(ξ) and gλ(z) =
∫
IR
g(z, t)eiλtdt.
Now, if g is also radial on IHn, which means that it depends only on |z| and
t, then it follows that the operators ĝ(λ) are diagonal on the Hermite basis for
L2(IRn).
The following functions are required in Theorem 2.6 below. For δ > −1, the
Laguerre functions of type δ are given by
Λδ
k(x) =
(
k!
(k + δ)!
)1/2
Lδ
k(x)e−
1
2
xx
δ
2 .
Also, for each λ > 0,
`λ
k(r) = (|λ|r2)
1−n
2 Λn−1
k (
1
2
|λ|r2), r ∈ IR+.
156 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
Theorem 2.6. If g ∈ L1(IHn) and g(z, t) = g0(|z|, t), then
ĝ(λ)hλ
α(x) = Cnµ(|α|, λ)hλ
α(x),
where
µ(k, λ) =
(
k!
(k + n− 1)!
)1/2
∞∫
0
gλ
0 (s)(
1
2
|λ|s2)
1−n
2 Λn−1
k (
1
2
|λ|s2)s2n−1ds,
and Cn is a constant which depends only on n.
P r o o f. It is clear that gλ(z) = gλ
0 (|z|) for some function gλ
0 . We can
therefore write
gλ
0 (r) =
∞∑
k=0
∞∫
0
gλ
0 (s)`λ
k(s)|λ|ns2n−1ds
`λ
k(r).
From this we see that we formally have
gλ(z) = Cn
∞∑
k=0
µ(k, λ)ϕλ
k(z),
where Cn = (2π)n21−n. It now follows that
gλ ∗λ ϕλ
k(z) = Cnµ(k, λ)ϕλ
k(z),
and hence from [7] we have that this formal Laguerre expansion in fact agrees
with the special Hermite expansion
gλ(z) =
∞∑
k=0
gλ ∗λ ϕλ
k(z) = Cn
∞∑
k=0
µ(k, λ)ϕλ
k(z). (2.7)
Now, since ĝ(λ) = Wλ(gλ), the theorem follows immediately from the last equa-
tion and Lemma 10 of [19].
We now consider a comparison of the algebras of radial and spherical functions
in what follows.
Let hn denote the (2n + 1)-dimensional Heisenberg algebra with generators
X1, . . . , Xn, U1, . . . , Un, Z
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 157
M.E. Egwe and U.N. Bassey
satisfying the commutation relations [Zj , Uj ] = Zj . We identify hn with IR2n+1 :=
IRn × IRn × IR. For this, let x = (x1, . . . , xn) and u = (u1, u2, . . . , un) denote the
canonical coordinates on IR2n+1. The map
p : IR2n+1 −→ hn : (x, u, ξ) 7→
∞∑
j=1
xjXj +
n∑
j=1
ujUjξZ
is a linear isomorphism providing suitable coordinates for hn, using the Mackev
basis.
We identify IHn with hn through the exponential map
exp : hn −→ IHn
with the usual group law and Haar measure dh in such a way that it coincides
with the product of Lebesgue measures, i.e.,
∫
IHn
f(h)dh =
∫
IR2n+1
f(x, u, ξ)dxdudξ.
Here, for (x, u, ξ) ∈ IHn, we have
(x, u, ξ)−1 = (−x,−u,−ξ).
The automorphisms are the dilations
δr(z, ξ) := (rz, r2ξ), z = (x, u).
For (x, u, ξ) ∈ IHn, define the Koranyi-norm by
|(x, u, ξ)| := (|x + iu|4 + 16ξ2)1/4 = ||x + iu|2 ± 14iξ|1/2.
This norm has the following properties:
(i) |δrg| = r|g| ∀ g ∈ IHn, r > 0,
(ii) |g| = 0 ⇔ g = 0,
(iii) |g−1| = |g|,
(iv) |g1g2| ≤ |g1|+ |g2| g1, g2 ∈ IHn.
In particular, | · | is a homogeneous norm and dK(g1, g2) := |g−1
1 g2| defines a
left-invariant metric on IHn.
R e m a r k 2.7. IHn, endowed with the Koranyi metric dk and the Haar mea-
sure, forms a space of homogeneous type in the sense of Coifman and Weiss [20].
158 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
In fact, denote by
Br(g) := {h ∈ IHn : |g−1h| < r}
the ball of radius r > 0 centred at g ∈ IHn. Then, by left-invariance and (i)
above, we have
|Br(g)| = |Br(0)| = |δr(B1(0))| = rQ|B1(0)|,
where Q = 2n + 2
is the homogeneous dimension of IHn.
Next, let U(hn) denote the universal enveloping algebra of hn and let the
Laplace element in U(hn) be given by
L :=
n∑
j=1
X2
j +
n∑
j=1
U2
j + Z2.
For X ∈ hn, we shall write X̃ for the left-invariant vector field on IHn, i.e.,
(X̃f)((h) =
d
dt
∣∣∣∣
t=0
f(h exp(tX))
for f a function on IHn which is differentiable at h ∈ IHn. Let ρ be the right
regular representation of IHn on L2(IHn), i.e.,
(ρ(h)f)(x) = f(xh)
for x, h ∈ IHn and f ∈ L2(IHn). If dρ is the derived representation, then we have
dρ(X) = X̃ for all X ∈ hn. In particular, if
∆IHn :=
n∑
i=1
X̃2
i +
n∑
i=1
Ũ2
i + Z̃
denotes the Laplacian on IHn, then dρ(L) = ∆IHn . Now set IR+ = (0,∞).
We have already seen that ∆IHn is not globally solvable. We now turn to the
Heisenberg heat equation defined on IHn × IR+ by
∂tU(h, t) = ∆U(h, t),
U(h, t) ∈ IHn × IR+. The fundamental solution of this equation is given by the
heat kernel Kt(h) which is obtained explicitly in [1] as
Kt(x, u, ξ) = cn
∫
IR
e−iλEe−tλ2
(
λ
sinhλt
)n
e−
1
4
λ(cot htλ)(x·x+u·u)dλ,
where cn = (4π)−n, λ ∈ IR∗ = IR \ {0}.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 159
M.E. Egwe and U.N. Bassey
Let ϕk
λ be the K-spherical function on IHn. That is, the distinguished spher-
ical function restricted to L1(K\G/K) where (K, G) is a Gelfand pair, K a
compact subgroup of Aut(IHn). In this case, G may be taken as a semi-direct
product of K and IHn (i.e., G := K n IHn) [10]. Thus ϕk
λ is a unique radial
function since it is a radial eigenfunction of ∆IHn [13, p. 38]. (In fact, elementary
spherical functions are radial functions [15]), i.e.,
ϕk
λ(u) = ψ(|u|).
Now rewriting the heat kernel, we have
Kt(h) = cn
∫
IRn
e−λξe−tλ2
ϕn(λt)e−
1
4
|h|2φ(λt)dλ
= cn
λ(ξ+λ2)∫
IRn
ϕn(λt)e−
1
4
|h|2φ(λt)dλ
= cnψλ(|h|, t)
which gives a radial function for K := U(n) and
Kt(h) = cnψλ(e−iθ|h|, t)
which gives a polyradial function for K := TTn. Applying dilations to the radial
function, we obtain
Kt(h) = δr(cnψλ(|h|, t)
= cnψλ(r|h|, r2t)
= cnt−n/2ϕn(h)δ−2
r (h)e|h|2/4t.
Let A be the subalgebra L1(IHn) (with respect to the right invariant Haar mea-
sure) generated by Kt, t > 0. We wish to state a lemma (Tauberian theorem)
which gives conditions, in terms of non-vanishing of transforms, for a closed ideal
I in L1(IHn) to be all the space L1(IHn).
First, we consider the spherical transform of any f ∈ L1(IHn). The Gelfand
spherical transform is defined for the commutative Banach algebra A as the
mapping from A to the continuous functions on its maximal ideal space M(A).
The maximal ideal space consists of all the non-zero continuous homomorphisms
from A to the complex numbers IC. As L1(K\G/K) is a commutative Banach
algebra, the spherical transform can be defined. Now the maximal ideal space
M(L1(K\G/K)) may also be expressed using the bounded spherical functions.
160 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
The set of bounded spherical functions consists of a Laguerre part and a Bessel
part. They are the following [14, 18]:
ϕλ
k(z, t) = e2πiλte−2π|λ||z|2
n∏
j=1
L
(0)
kj
(4π|λ||zj |2), λ ∈ IR∗, k ∈ (ZZ+)n,
J ρ
0 =
n∏
j=1
J0(ρj |zj |), ρ ∈ (IR+)n,
respectively. Here L
(0)
k is the Laguerre polynomial of degree k and J0 is the Bessel
function (of the first kind) of index 0. The spherical transform of a function is
then given by
f̃(λ; k) =
∫
IHn
f(z, t)ϕλ
k(z, t) dzdt,
f̃(0; ρ) =
∫
IHn
f(z, t)J ρ
0 (z) dzdt.
Definition 2.8. Let A be an algebra. (Here, an Ideal of A is always a two-
sided ideal.) The primitive ideal space of A, denoted by Prim(A), is the space of
all ideals I of A of the form I = Ker(T ), where T (V ) denotes an algebraically
irreducible representation of A on a vector space V . We provide Prim(A) with
the Jacobson topology. In this topology, a subset C of Prim(A) is closed if it is
the hull H(I) of some ideal I of A, i.e., if
C = H(I) = {J ∈ Prim(A) : J ⊃ I}.
For a subset C ⊂ Prim(A), let
Ker(C) =
⋂
j∈C
J ⊂ A and I(C) =
⋂
H(I)=C
I.
The hull of I(C) contains C.
For certain algebras A, we have H(I(C)) = C, i.e., there exists a minimal
ideal j(C) with hull C. That means there exists an ideal j(C) of A such that
the hull of j(C) is equal to C and j(C) ⊂ I for every ideal I of A whose hull is
contained in C.
R e m a r k 2.9. It was shown in [5] that j(C) exists for every closed subset
C in the primitive ideal space for the Schwartz algebra of a nilpotent Lie group.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 161
M.E. Egwe and U.N. Bassey
Lemma 2.10. Let II ⊂ L1(IHn) be a closed ideal such that
(i) for each (λ, k) ∈ IR∗ × (ZZ+)n, there exists f ∈ II such that
f̃(λ; k) 6= 0,
(ii) for each ρ ∈ (IR+)n, there exists f ∈ II such that
f̃(0, ρ) 6= 0.
Then II = L1(IHn).
P r o o f. Assume without loss of generality that f ∈ S(IHn). This is possible
since S(IHn) is dense in L1(IHn). Now, by hypothesis, II is closed and therefore
must be the hull of some ideal, say, J of L1(IHn). This makes II a subset of
Prim(L1(IHn)) since for any f spherical, ϕλ
k(0) = f̃(λ, k) 6= 0, and f̃(0, ρ) 6= 0.
Thus II = H(I) = {M ∈ Prim(L1(IHn) : M ⊃ J}.
Now, since IHn is a nilpotent Lie group, it follows from Remark 2.9 that
H(I(II)) = II,
=⇒ L1(IHn) = II
since II is a closed ideal.
Theorem 2.11. Let Ar(IHn) and Sp(IHn) denote the algebras of radial and
spherical functions on IHn, respectively. Define an operator
T : Sp(IHn) −→ Ar(IHn) by
T (ϕ) = Cnϕk
λ(u)δr(u)e|u|
2/4e−iλt, u ∈ IHn
= Cnϕk
λ(|u|, t).
Then T is an algebraic isomorphism of Ar(IHn) and Sp(IHn).
P r o o f. First recall that the heat equation on IRn is given by
ut(t, x) = ∆u(t, x),
u(0, x) = δ(x).
Now the calculation of the Gaussian integral
u(ε, x) = 2π−n
∫
IRn
eix·ξ−ε|ξ|2dξ
162 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
On Isomorphism Between Certain Group Algebras on the Heisenberg Group
gives explicitly the fundamental solution of the heat equation as [20, p. 289]
et∆δ(x) = (4πt)−n/2e−|x|
2/4t, t > 0, x ∈ IRn.
Thus we have the heat kernel qt(x) on IRn as given above. Now, let V denote the
vector space of all linear combinations of qt, t > 0. By the formula in the theorem,
T restricted to V is an isomorphism of algebras. Moreover, for all f ∈ V , we have
∫
Sp(IHn)
Tϕ =
∫
Ar(IHn)
ϕ.
On the other hand, if we denote E the space L1
rad(IR
n, eC|x|dx) for sufficiently
large C, then T is continuous from E to Sp(IHn) ⊂ L1(IHn). Since V is dense in
E, it follows that T (E) ⊂ L1(K\IHn/K) and for all ϕ ∈ E, we have
∫
L1(K\IHn/K)
Tϕ =
∫
IRn
ϕ.
From [15], any f ∈ E can be decomposed into its positive and negative parts
with each component belonging to E. Thus decomposing ϕ yields
‖ϕ‖L1
rad
=
∫
IRn
ϕ+ +
∫
IRn
ϕ− =
∫
Sp(IHn)
Tϕ+ +
∫
Sp(IHn)
Tϕ− = ‖Tϕ‖L1(IHn)
showing that the closure of T |V is an isometry of L1
rad(IR
n) with Sp(IHn) and this
closure is equal to T. Hence the proof follows by Lemma 2.10.
Acknowledgement. The authors quite appreciate the complete and con-
structive suggestions of the referee on the center of the Heisenberg group which
was initially assumed intrinsic.
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