On Wigner transforms in infinite dimensions

0.23 MB·English

by  Ingrid Beltita

ON WIGNER TRANSFORMS IN INFINITE DIMENSIONS
5 INGRIDBELTIT¸A˘,DANIELBELTIT¸A˘,ANDMARIUSMA˘NTOIU
1
0
Abstract. WeinvestigatetheSchro¨dingerrepresentationsofcertaininfinite-
2
dimensionalHeisenberggroups,usingtheircorrespondingWignertransforms.

n
a
J

2
2 Contents

] 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
T 2. Preliminaries on Fourier transforms on uniform spaces . . . . . . . . . 2
R 3. Operator calculus on topological groups . . . . . . . . . . . . . . . . . 4
. 4. Some computations involving Gaussian measures . . . . . . . . . . . . 7
h
t 5. Infinite-dimensional Heisenberg groups and Wigner transforms . . . . 10
a
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
m
[

1
v 1. Introduction
4
0 The topic of this paper belongs to representation theory of Heisenberg groups,
4 more specifically we investigate to what extent the square-integrability properties
5 of the Schr¨odinger representations carry over to the setting of infinite-dimensional
0 Heisenberg groups. Recall that the Schr¨odinger representation of the (2n + 1)-
1. dimensional Heisenberg group H is a group representation
2n+1
0
π: H →B(L2(Rn,dλ))
5 2n+1
1
and its corresponding Wigner transform is a unitary operator
:
v W: L2(Rn,dλ)⊗¯L2(Rn,dλ)→L2(Rn×Rn,dµ)
i
X
wherewedenotebydλtheLebesguemeasureonRn,byB(L2(Rn,dλ))thebounded
ar linear operators on the complex Hilbert space L2(Rn,dλ), and by dµ a suitable
normalization of the measure dλ⊗dλ on Rn×Rn. One way to define the Wigner
transformisthatofdefiningW(f⊗ϕ)∈L2(Rn×Rn,dµ)asaFouriertransformof
the representation coefficient (π(·)f |ϕ)|Rn×Rn×{0} ∈L2(Rn×Rn×{0},dλ⊗dλ),
recalling that H =Rn×Rn×R as smooth manifolds.
2n+1
As the translation invariance property of the Lebesgue measure plays a central
role in the above discussion, it is not straightforward to replace here Rn by an
Date:January22,2015.
2000 Mathematics Subject Classification. Primary22E66;Secondary 28C05,28C20,22E70.
Keywords andphrases. Wignertransform;Gaussianmeasure;infinite-dimensionalLiegroup.
The research of I. Belti¸t˘a and D. Belti¸t˘a has been partially supported by the Grant of the
RomanianNational AuthorityforScientific Research, CNCS-UEFISCDI, projectnumber PN-II-
ID-PCE-2011-3-0131. M.M˘antoiuhas beensupportedbytheFondecyt Project1120300 andthe
Nu´cleoMileniodeF´ısicaMatema´ticaRC120002.
1
2 INGRIDBELTIT¸A˘,DANIELBELTIT¸A˘,ANDMARIUSMA˘NTOIU
infinite-dimensional real Hilbert space. It is customary in the infinite-dimensional
analysisto try toreplacethe Lebesguemeasureby aGaussianmeasure,andthis is
whatwewilldointhe presentpaperaswell. Furthermore,the mainproblemsthat
we must address are to construct the Wigner transform as a unitary operator on
square-integrablefunctions of infinitely many variables and to realize the image of
that unitary operator as an L2-space, which amounts to determining the infinite-
dimensionalanalogueofthe measuredµ fromthe above paragraph. In some sense,
theseproblemsformacomplementtotheonesaddressedinourrecentinvestigation
of square-integrable families of operators [BBM14].
The present paper is organized as follows. Sections 2 and 3 develop an abstract
frameworkforthestudyofWignertransformsassociatedtounitaryrepresentations
of general topological groups. The main ingredients of that framework are the
Fourier transforms on uniform spaces and an operator calculus that involves the
Banach algebra structures of the dual of the spaces of left uniformly continuous
functions on topological groups. Then Section 4 records some computations with
Gaussian functions and their Wigner transforms. In Section 5 we introduce the
infinite-dimensional Heisenberg groups and their Schr¨odinger representations for
which we construct their corresponding Wigner transforms in Theorem 5.6.
Throughout this paper we denote by ⊗¯ the Hilbertian scalar product and by
B(X) and X′ the spaces of all bounded linear operators and bounded linear func-
tionals on some Banach space X, respectively, and it will always be clear from the
context if the ground field is R or C.
2. Preliminaries on Fourier transforms on uniform spaces
Let X be any Hausdorff uniform space. We denote by UC and UC the vari-
b
ous spaces of uniformly continuous functions and uniformly continuous bounded
functions on any uniform space, respectively. Note that UC (X):=UC (X,C) is a
b b
Banachspace,sowemayconsiderits dualBanachspaceM(X):=UC (X)′, which
b
should be thought of as a space of generalized complex measures on X.
Definition 2.1. We define the uniform space dual to X as
X∇ :=UC(X,R)
endowed with the uniform structure of pointwise convergence.
Remark 2.2. There exists a natural injective mapping
η : X →(X∇)∇, x7→η
X x
where η (f) = f(x) for every f ∈ UC(X,R) and x ∈ X. It is easily seen that η is
x
uniformly continuous.
Definition 2.3. The Fourier transform on X is the linear mapping
F: M(X)→ℓ∞(X∇), (Fµ)(f)=hµ,eifi
for f ∈X∇ and µ∈M(X). We will also denote µ:=Fµ for µ∈M(X).
Note that F is a bounded linear mapping and in fact kFk≤1.
b
Proposition 2.4. The Fourier transform F: M(X)→ℓ∞(X∇) is injective.
ON WIGNER TRANSFORMS IN INFINITE DIMENSIONS 3
Proof. Let µ ∈ M(X) with Fµ = 0. In order to prove that µ = 0, it suffices to
show that for an arbitrary real-valued function f ∈ UC (X) we have hµ,fi = 0.
b
First recall that we have
eitr−1
lim −ir=0
t→0 t
uniformly for r in any compact subset of R. Therefore
eitf −1
lim =if
t→0 t
in UC (X), hence
b
(Fµ)(tf)−(Fµ)(0)
ihµ,fi= lim =0,
t→0 t
and we are done. (cid:3)
Lemma 2.5. If µ∈M+(X) then the following assertions hold:
(1) We have kFµk ≤hµ,1i.
∞
(2) For all f,h∈X∇ we have
|(Fµ)(f)−(Fµ)(h)|2 ≤2hµ,1i(hµ,1i−Re(Fµ)(f −h)).
Proof. For Assertion (1) recall from Definition 2.3 that kFk≤1, hence
kFµk ≤kµk=hµ,1i
∞
wherethelatterequalityfollowssinceµ: UC (X)→Cisapositivelinearfunctional
b
on the C∗-algebra UC (X).
b
To prove Assertion (2), let f,h∈X∇ arbitrary. By using the Cauchy-Schwartz
inequality we get
|(Fµ)(f)−(Fµ)(h)|2 =|hµ,eif −eihi|2
≤hµ,|eif −eih|2ihµ,12i
=hµ,1ihµ,2−2Re(ei(f−h))i
=2hµ,1i(hµ,1i−hµ,Re(ei(f−h))i)
=2hµ,1i(hµ,1i−Re((Fµ)(f −h))),
where we also used the fact that |eit−eis|2 =2−2Re(ei(t−s)) for all t,s∈R. (cid:3)
Lemma 2.6. Let X and Y be uniform spaces. Assume that D is a dense subset of
X and A is a uniformly equi-continuous family of mappings from X into Y. Then
the following uniform structures on the mappings from X into Y induce the same
uniform structure on A:
(a) the structure of uniform convergence on the precompact subsets of X;
(b) the structure of pointwise convergence on X;
(c) the structure of pointwise convergence on D.
Proof. SeetheproofoftheAscoli-Arzel`atheoremonuniformspacesin[Bou69]. (cid:3)
We now introduce the linear space of tight measures M (X) on a uniform
t
space X (see [Pa13, Sect. 5.1] for more information in this connection). Namely,
M (X) is the set of all linear functionals ϕ: UC (X) → C with the property
t b
4 INGRIDBELTIT¸A˘,DANIELBELTIT¸A˘,ANDMARIUSMA˘NTOIU
that for every net {f } in UC (X) with supkf k < ∞ and limf = 0 uni-
i i∈I b i ∞ i
i∈I i∈I
formly on every compact subset of X one has limϕ(f ) = 0. We also denote
i
i∈I
M +(X):=M (X)∩M+(X).
t t
Lemma 2.7. Let X be a uniform space and ϕ: UC (X)→ C a self-adjoint linear
b
functional with its positive part and negative part ϕ±. Then ϕ ∈ Mt(X) if and
only if ϕ±Mt(X).
Proof. See [F¨eo67]. (cid:3)
Proposition 2.8. The following assertions hold:
(1) Let µ ∈ M+(X). We have Fµ ∈ UC (X∇) if and only if Re(Fµ) is
b
continuous at 0∈X∇.
(2) If µ∈Mt(X), then Fµ∈UCb(X∇).
Proof. Assertion (1) follows at once by Lemma 2.5.
For proving Assertion (2), we see from Lemma 2.7 that we may assume µ ∈
M +(X). Then, according to Assertion (1), it suffices to show that for every
t
µ ∈ M +(X) the function Re(Fµ): X∇ → C is continuous at 0 ∈ X∇. To this
t
end, let us assume that limf = 0 in X∇. In other words, {f } is a net of
j j j∈J
j∈J
uniformly continuous real functions on X with limf = 0 pointwise on X. By
j
j∈J
using Lemma 2.6, we see that {cosf } is a uniformly bounded net in UC (X)
j j∈J b
which converges to 1 ∈ UC (X) uniformly on the compact subsets of X. Since
b
µ∈M (X), we then get
t
limRe(Fµ)(f )=limRehµ,eifji=limhµ,cosf i=1.
j j
j∈J j∈J j∈J
Therefore the function Re(Fµ) is continuous at 0 ∈ X∇, and this completes the
proof. (cid:3)
Remark 2.9. Lemma 2.5 and Proposition 2.8 are straightforward extensions of
some results from [Bou69, §6, no. 8].
3. Operator calculus on topological groups
In this section we introduce an operator calculus for unitary representations of
topologicalgroups,sinceitwillallowustohandleinSection5somerepresentations
ofinfinite-dimensionalHeisenberggroups,whichareLiegroupsmodeledonHilbert
spaces. In the case of Banach-Lie groups, the space of continuous 1-parameter
subgroups and exponential map, as defined below, agree with the usual notions
of Lie algebra and exponential map from the Lie theoretic setting (see [Nee06] for
extensive information in this connection). We also note that in the case of finite-
dimensional nilpotent Lie groups the present operator calculus recovers the Weyl
calculus used in [GH14] in the study of L´evy processes.
Let G be a topological group endowed with the right uniform structure. We
recall that a basis of this uniform structure is provided by the sets
Sλ ={(x,y)∈G×G|yx−1 ∈V},
V
where V ∈V (1). Consider the corresponding space of mappings
G
L(G)={X: R→G|X homomorphism of topological groups}
ON WIGNER TRANSFORMS IN INFINITE DIMENSIONS 5
with the structure of uniform convergence on the compact subsets of R. Hence a
basis of this uniform structure consists of the entourages
Sλ ={(X,Y)∈L(G)×L(G)|(∀t∈[−n,n]) Y(t)X(t)−1 ∈V}
n,V
parameterized by V ∈V (1) and n∈N (see [HM07, Def. 2.6]).
G
It is easily seen that the exponential mapping
exp : L(G)→G, exp X :=X(1)
G G
is uniformly continuous, hence it gives rise to a unital ∗-homomorphism of C∗-
algebras
UC (exp ): LUC (G)→UC (L(G)), f 7→f ◦exp
b G b b G
with the dual map
UC ′(exp ): (UC (L(G)))′ →(LUC (G))′.
b G b b
We recall from [Gru] (see also [B11, Th. 3.9]) that (LUC (G))′ is a unital associa-
b
tive Banach algebra in a natural way and every continuous unitary representation
π: G→B(H ) gives rise to a Banach algebra representation
π
π : (LUC (G))′ →B(H)
L b
such that (πL(ν)φ | ψ) = hbν,(π(·)φ | ψ)i for φ,ψ ∈ H and ν ∈ (LUCb(G))′.
Therefore we get a bounded linear operator
b
π : (UC (L(G)))′ →B(H)
L b
such that the diagram e
(LUC (G))′ πbL // B(H)
UCb′(expG)bOO qqqqqπeqqLqqq88
(UC (L(G)))′
b
is commutative.
Setting 3.1. Throughout the rest of this section we fix a continuous unitary rep-
resentationπ: G→B(H)ofthe abovetopologicalgroupGonthe complex Hilbert
space H and we assume the setting defined by the following data:
• a uniform space Ξ and a uniformly continuous map θ: Ξ→L(G),
• a locally convex space Γ such that there exists an injective continuous
inclusion map Γ֒→M(L(G)∇),
• a locally convex space H such that there exists an injective continuous
Ξ,∞
inclusion map H ֒→H.
Ξ,∞
Also let ηL(G): L(G) → (L(G)∇)∇ be the uniformly continuous map defined in
Remark 2.2.
Definition 3.2. We say that Γ and θ are compatible if the linear mapping
FΞ: Γ→UCb(Ξ), µ7→µ◦ηL(G)◦θ
is well defined and injective. b
If this is the case, then we denote Q := F (Γ) ֒→ UC (Ξ) and endow it with
Ξ Ξ b
the topology which makes the Fourier transform
F : Γ→Q
Ξ Ξ
6 INGRIDBELTIT¸A˘,DANIELBELTIT¸A˘,ANDMARIUSMA˘NTOIU
into a linear toplogical isomorphism. We then also have the linear toplogical iso-
morphism (F′)−1: Γ′ →Q′ .
Ξ Ξ
Lemma 3.3. If Γ and θ are compatible, then the following conditions are equiva-
lent:
(1) We have the well-defined continuous sesquilinear mapping
Aπ,θ: H ×H →Q , (φ,ψ)7→Aπ,θφ:=(π(exp (θ(·)))φ|ψ).
Ξ,∞ Ξ,∞ Ξ ψ G
(2) There exists a unique continuous sesquilinear mapping
W: H ×H →Γ,
Ξ,∞ Ξ,∞
suchthatfor allφ,ψ ∈H wehave F (W(φ,ψ))=(π(exp (θ(·)))φ|ψ).
Ξ,∞ Ξ G
Proof. Ifcondition(2)issatisfied,thenAπ,θ =F ◦W,hencecondition(1)follows
Ξ
sinceF : Γ→Q isa lineartopologicalisomorphism. Forthe samereason,italso
Ξ Ξ
follows that if condition (1) holds true, then (2) is satisfied. (cid:3)
Definition3.4. AssumethatΓandθarecompatibleandtheequivalentconditions
in Lemma 3.3 are satisfied. Then the sesquilinear map W is called the Wigner
transform. The operator calculus for π along θ is the linear map
Opθ: Γ′ →L(H ,H′ )
Ξ,∞ Ξ,∞
defined by
(Opθ(a)φ|ψ)=h(F′)−1(a),(π(exp (θ(·)))φ|ψ)i (3.1)
Ξ G
∈Q′Ξ ∈QΞ
| {z } | {z }
fora∈Γ′ andφ,ψ ∈H ,whereH′ denotesthespaceofcontinuousantilinear
Ξ,∞ Ξ,∞
functionals on H .
Ξ,∞
Remark3.5. InthesettingofDefinition3.4wehaveforalla∈Γ′andφ,ψ ∈H
Ξ,∞
(Opθ(a)φ|ψ)=h(F′)−1(a),Aπ,θφi=ha,F−1(Aπ,θφ)i=ha,W(φ,ψ)i,
Ξ ψ Ξ ψ
where the later equality follows by Lemma 3.3(2).
Definition 3.6. Assume the setting of Definition 3.4. We say that the represen-
tationπ satisfies the orthogonality relations along the mapping θ: Ξ→L(G) if the
following conditions are satisfied:
(1) The linear subspace H is dense in H.
∞,Ξ
(2) There exists a continuous, positive definite, sesquilinear, inner product on
ΓsuchthatifwedenotebyΓ thecorrespondingHilbertspaceobtainedby
2
completion, then the sesquilinear mapping W: H ×H →Γ extends
Ξ,∞ Ξ,∞
to a unitary operator
W: H⊗¯H→Γ (3.2)
2
which is still called the Wigner transform.
Remark 3.7. InDefinition3.6,sincetheinnerproductonΓiscontinuous,itgives
risetoacontinuousinjectivemapΓ ֒→Γ′. Byusingthefactthattheoperator(3.2)
2
is an isometry, we easily get
(∀φ,ψ ∈H) Opθ(W(φ,ψ))=(·|ψ)φ.
ON WIGNER TRANSFORMS IN INFINITE DIMENSIONS 7
4. Some computations involving Gaussian measures
This section records some auxiliary facts that will be needed in the proof of
Theorem 5.6.
Notation 4.1. We shall use the following notation:
(1) For x = (x ,...,x ) ∈ Rm we set x′ = (x ,...,x ) ∈ Rm−1 if m ≥ 2,
1 m 1 m−1
hence
x=(x′,x )∈Rm−1×R.
m
(2) For every t>0 we denote
(∀x∈R) γt(x)= 1 1/2e−x22t,
(cid:16)2πt(cid:17)
so that γ (x)dx is the centered Gaussian probability measure on R with
t
variance t (and mean 0).
Remark 4.2. We recall that
Z γt(x)eivxdx=(cid:16)2tπ(cid:17)1/2γ1/t(v)=e−tv22 (4.1)
R
for all v ∈R and t>0.
Remark 4.3. For any integer m ≥ 1 recall the unitary operator that gives the
integral kernels of operators obtained by classical Weyl calculus with L2-symbols
T : L2(Rm×(Rm)′)→L2(Rm×Rm)
m
defined by
1 x+y
(T (a))(x,y)= a( ,ξ)eihx−y,ξidξ,
m (2π)m/2 Z 2
Rm
with its inverse
T−1: L2(Rm×Rm)→L2(Rm×(Rm)′)
m
given by
1 v v
(T−1(K))(x,ξ)= K(x+ ,x− )e−ihv,ξidv, (4.2)
m (2π)m/2 Z 2 2
Rm
for every m≥1.
Then for arbitrary m ,m ≥1 there exist the natural unitary operators
1 2
V : L2(Rm1 ×Rm1)⊗¯L2(Rm2 ×Rm2)→L2(Rm1+m2 ×Rm1+m2)
given by
(V(f ⊗f ))((x ,x ),(y ,y ))=f (x ,y )f (x ,y )
1 2 1 2 1 2 1 1 1 2 2 2
for xj,yj ∈Rmj, fj ∈L2(Rmj ×Rmj), j =1,2, and
W: L2(Rm1 ×(Rm1)′)⊗¯L2(Rm2 ×(Rm2)′)→L2(Rm1+m2 ×(Rm1+m2)′)
given by
(W(g ⊗g ))((x ,x ),(ξ ,ξ ))=g (x ,ξ )g (x ,ξ )
1 2 1 2 1 2 1 1 1 2 2 2
8 INGRIDBELTIT¸A˘,DANIELBELTIT¸A˘,ANDMARIUSMA˘NTOIU
for xj ∈Rmj, ξj ∈L2((Rmj)′), fj ∈L2(Rmj ×(Rmj)′), j =1,2, and the diagram
L2(Rm1 ×(Rm1)′)⊗¯L2(Rm2 ×(Rm2)′) W // L2(Rm1+m2 ×(Rm1+m2)′)
Tm1⊗Tm2 Tm1+m2
(cid:15)(cid:15) (cid:15)(cid:15)
L2(Rm1 ×Rm1)⊗¯L2(Rm2 ×Rm2) V //L2(Rm1+m2 ×Rm1+m2)
is commutative.
The following simple lemma plays a key role in the presentpaper since it allows
us to determine integral kernels of operators on Gaussian L2-spaces in terms of
integral kernels on Lebesgue L2-spaces.
Lemma 4.4. The unitary operator T : L2(R2)→L2(R2) defined by
1
1 x+y
(∀a∈L2(R2)) (T (a))(x,y)= a( ,ξ)ei(x−y)ξdξ
1 (2π)1/2 Z 2
R
has the property
(∀t>0) T−1((γ ⊗γ )1/2)=(γ ⊗γ )1/2.
1 t t t/2 1/8t
Proof. Let us denote K =(γ ⊗γ )1/2. Then we have
t t
K(x,y)= 1 1/2e−x24+ty2
(cid:16)2πt(cid:17)
hence, using (4.2), one obtains
(T−1(K))(x,ξ)= 1 e−41t (x+v2)2+(x−v2)2 e−ivξdv
1 2πt1/2 Z (cid:0) (cid:1)
R
= 1 e−x22t e−v82te−ivξdv
2πt1/2 Z
R
= 2πt11/2 e−x22t(8πt)1/2Z γ4t(v)eivξdv
R
= 2 1/2e−x22te−2tξ2
(cid:16)π(cid:17)
= 1 1/2e−xt2 4t 1/2e−4tξ2 1/2
(cid:16)(cid:16)πt(cid:17) (cid:16)π(cid:17) (cid:17)
=(γ (x)γ (ξ))1/2
t/2 1/8t
wherethethirdfromlastequalityfollowsby(4.1),andthiscompletestheproof. (cid:3)
Proposition 4.5. Let t ≥t ≥···>0 and for m=1,2,... define
1 2
γ : Rm →R, γ (x ,...,x ):=γ (x )···γ (x )
m m 1 m t1 1 tm m
e e
ON WIGNER TRANSFORMS IN INFINITE DIMENSIONS 9
and dγ :=γ (x)dx. Then for m≥2 the diagram
m m
e eL2(Rm−1×(Rm−1)′) β //L2(Rm×(Rm)′)
Sm−1 Sm
(cid:15)(cid:15) (cid:15)(cid:15)
L2(Rm−1×(Rm−1)′) α // L2(Rm×(Rm)′)
Tm−1 Tm
(cid:15)(cid:15) (cid:15)(cid:15)
L2(Rm−1×Rm−1) η // L2(Rm×Rm)
Um−1 Um
(cid:15)(cid:15) (cid:15)(cid:15)
L2(Rm−1×Rm−1,dγ ×dγ ) ι //L2(Rm×Rm,dγ ×dγ )
m−1 m−1 m m
is commutative, where the veertical arerows are unitary operators deefinedeby
1 x x ξ ξ
1 ℓ 1 ℓ
(S (b))(x ,...,x ,ξ ,...,ξ )= ·b( ,..., , ,..., )
ℓ 1 ℓ 1 ℓ (t ···t )2 t t t t
1 ℓ 1 ℓ 1 ℓ
1 x+y
(T (a))(x,y)= a( ,ξ)eihx−y,ξidξ,
ℓ (2π)l/2 Z 2
Rl
(U (K))(v,w)=K(v,w)γ (v)−1/2γ (w)−1/2,
ℓ ℓ ℓ
for ℓ∈{m−1,m} and the horizontal arrows aere isometeries defined by
(β(b))(x,ξ)=b(x′,ξ′)γ (x )1/2γ (ξ )1/2
1/2 m 1/8t2 m
m
(α(a))(x,ξ)=a(x′,ξ′)γ (x )1/2γ (ξ )1/2
tm/2 m 1/8tm m
(η(K))(x,y)=K(x′,y′)γ (x )1/2γ (y )1/2
tm m tm m
(ι(Q))(v,w)=Q(v′,w′).
Proof. It is clear that the horizontal arrows in the diagram are isometries, while
the vertical arrows are unitary operators.
Moreover,for K ∈L2(Rm−1×Rm−1) we have
(U (η(K)))(v,w) =(η(K))(v,w)γ (v)−1/2γ (w)−1/2
m m m
=K(v′,w′)γtme(vm)1/2γtme(wm)1/2γm(v)−1/2γm(w)−1/2
=K(v′,w′)γm−1(v′)−1/2γm−1(w′)−e1/2 e
=(U (K))(v′,w′)
m−1 e e
=(ι(U (K)))(v,w),
m−1
hence the lower part of the diagram in the statement is commutative. As regards
the middle part of the diagram, we have
(η(T (a)))(x,y)=(T (a))(x′,y′)γ (x )1/2γ (y )1/2
m−1 m−1 tm m tm m
=(T (a))(x′,y′)·(T ((γ ⊗γ )1/2))(x ,y )
m−1 1 tm/2 1/8tm m m
=(T (a⊗(γ ⊗γ )1/2))(x′,x ,y′,y )
m tm/2 1/8tm m m
=(T (α(a)))(x,y)
m
where the second equality follows by Lemma 4.4, while the third equality is a
consequence of Remark 4.3.
10 INGRIDBELTIT¸A˘,DANIELBELTIT¸A˘,ANDMARIUSMA˘NTOIU
Finally, by using the fact that
(∀t,a>0)(∀x∈R) a1/2γ (ax)=γ (x),
t t/a
it follows by a straightforwardcomputation that the upper part of the diagram in
the statement is commutative. (cid:3)
Remark 4.6. In order to point out the role of the unitary operators
U : L2(Rℓ×Rℓ)→L2(Rℓ×Rℓ,dγ ×dγ )
ℓ ℓ ℓ
in Proposition 4.5, we note the following fact. The mueltipliceation operator
V : L2(Rℓ)→L2(Rℓ,dγ ), V f =f(γ )−1/2
ℓ ℓ ℓ ℓ
is unitary. Furthermore, for any K ∈L2(eRℓ×Rℓ), if weedenote the corresponding
integral operator by
A : L2(Rℓ)→L2(Rℓ), (A f)(x)= K(x,y)f(y)dy,
K K Z
Rℓ
then V A V−1: L2(Rℓ,dγ ) → L2(Rℓ,dγ ) is the integral operator whose integral
ℓ K ℓ ℓ ℓ
kernel is U (K)∈L2(Rℓ×Rℓ,dγ ×dγ ).
ℓ ℓ ℓ
e e
5. Infinite-dimensional Heeisenbeerg groups and Wigner transforms
In this section we study representations of a special type of infinite-dimensional
Heisenberggroups,asintroducedinthefollowingdefinition. Amoregeneralframe-
work can be found for instance in [Nee00] and [DG08].
For the sake of completeness we recall here a few facts from [BB10, Sect. 3].
Definition 5.1. If V is a real Hilbert space, A ∈ B(V) with (Ax | y) = (x | Ay)
for all x,y ∈V, and moreoverKerA={0},then the Heisenberg algebra associated
with the pair (V,A) is the real Hilbert space h(V,A) = V ∔V ∔R endowed with
the Lie bracket defined by [(x ,y ,t ),(x ,y ,t )] =(0,0,(Ax | y )−(Ax | y )).
1 1 1 2 2 2 1 2 2 1
The correspondingHeisenberg group H(V,A)=(h(V,A),∗) is the Lie groupwhose
underlying manifold is h(V,A) and whose multiplication is defined by
(x ,y ,t )∗(x ,y ,t )=(x +x ,y +y ,t +t +((Ax |y )−(Ax |y ))/2)
1 1 1 2 2 2 1 2 1 2 1 2 1 2 2 1
for (x ,y ,t ),(x ,y ,t )∈H(V,A).
1 1 1 2 2 2
Gaussian measures and Schro¨dinger representations. Let V be a realHil-
−
bertspacewiththescalarproductdenotedby(·|·) . Foreveryvectora∈V and
− −
every symmetric, nonnegative, injective, trace-class operator K on V there exists
−
a unique probability Borel measure γ on V such that
−
(∀x∈V−) Z ei(x|y)−dγ(y)=ei(a|x)−−12(Kx|x)−
V−
(see for instance [Kuo75, Th. I.2.3]). We also have
a= y dγ(y) and Kx= (x|y) ·(y−a)dγ(y) for all x∈V ,
Z Z − −
V− V−
where the integrals are weakly convergent, and γ is called the Gaussian measure
with the mean a and the variance K.