Geometric representation theory for unitary groups of operator algebras

0.25 MB·English

by  Daniel Beltiţ\ua

Geometric representation theory for unitary groups
5
0
of operator algebras
0
2

n
Daniel Belti¸t˘a1 and Tudor S. Ratiu2
a
J

5 February 1, 2008

]
T
R Abstract
.
h Geometric realizations for the restrictions of GNS representations to unitary groups of
t ∗
a C -algebras are constructed. These geometric realizations use an appropriate concept of
m reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic
[ sections,aclassofcomplexcoadjointorbitsofthe correspondingrealBanach-Liegroupsare
described and some homogeneous holomorphic Hermitian vector bundles that are naturally
1
associated with the coadjoint orbits are constructed.
v
Keywords: geometric realization; homogeneous vector bundle; reproducing kernel; oper-
7
5 ator algebra
0 MSC 2000: Primary 22E46; Secondary 46L30; 22E46;58B12;46E22
1
0
5 1 Introduction
0
/
h The study of geometric properties of state spaces is a basic topic in the theory of operator
t
a algebras (see, e.g., [2] and [3]). The GNS construction produces representations of operator
m
algebrasoutofstates. Fromthispointofview,wethinkitinterestingtoinvestigatethegeometry
:
v behind these representations.
i One method to do this is to proceed as in the theory of geometric realizations of Lie group
X
representations (see e.g., [20], [22], [29], [23]) and to try to build the representation spaces as
r
a spaces of sections of certain vector bundles. The basic ingredient in this construction is the
reproducing kernel Hilbert space (see, e.g., [6], [35], [25], [33], [7], [11], [29]).
In the present paper we show that the aforementioned method can indeed be applied to
the case of group representations obtained by restricting GNS representations to unitary groups
of C∗-algebras. More precisely, for these representations, we construct one-to-one intertwining
operators from the representation spaces onto reproducing kernel Hilbert spaces of sections of
certain Hermitian vector bundles (Theorem 5.4). The construction of these vector bundles is
based on a choice of a sub-C∗-algebra that is related in a suitable way to the state involved in
the GNS construction (see Construction 3.1). It turns out that, in the case of normal states
of W∗-algebras, there is a natural choice of the subalgebra (namely the centralizer subalgebra),
and the base of the corresponding vector bundle is just one of the symplectic leaves studied in
1Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700 Bucharest, Romania.
[email protected].
2Section de Math´ematiques and Centre Bernoulli, E´cole Polytechnique F´ed´erale de Lausanne, CH-1015 Lau-
sanne, Switzerland. [email protected].
1
Belti¸t˘a and Ratiu: Geometric representations 2
our previous paper [10]. Since the corresponding symplectic leaves are just unitary orbits of
states, the geometric representation theory initiated in thepresent paperprovides, in particular,
a geometric interpretation of the result in [24], namely the equivalence class of an irreducible
GNS representation only depends on the unitary orbit of the corresponding pure state.
In [16] and references therein one can find several interesting results regarding the classifi-
cation of unitary group representations of various operator algebras. The point of the present
paper is to show that some of these representations (namely the ones obtained by restricting
GNSrepresentationstounitarygroups)canberealizedgeometrically followingthepatternofthe
classical Borel-Weil theorem for compact groups. This raises the challenging problem of finding
geometric realizations of more general representations of unitary groups of operator algebras.
Similar results for other classes of infinite-dimensional groups have been already obtained: see
[28], [21], [39] for direct limit groups, and [14], [30], [31] for groups related to operator ideals.
The same problem of geometric realizations for representations of the restricted unitary group
was raised at the end of [15].
Thestructureof thepaperis as follows. Sincethereproducingkernels weneed inthepresent
paper show up most naturally in a C∗-algebraic setting (Construction 5.1), we establish in Sec-
tion 2 the appropriate versions of a number of results in [10]. Section 3 gives a general construc-
tion of homogeneous Hermitian vector bundles associated with GNS representations. Section 4
is devoted to the concept of reproducing kernel suitable for the applications we have in mind.
In Section 5 we construct such reproducing kernels out of GNS representations and we prove
our main theorems on geometric realizations of GNS representations (Theorems 5.4 and 5.8).
2 Coadjoint orbits and C∗-algebras with finite traces
In this section we extend to a C∗-algebraic framework a number of results that were proved in
[10] for symplectic leaves in preduals of W∗-algebras. We begin by establishing some notation
that will be used throughout the paper.
Notation 2.1 For a unital C∗-algebra A with the unit 1 we shall use the following notation:
{a}′ = {b ∈ A| ab= ba} whenever a∈ A,
Aϕ = {a ∈ A| (∀b ∈ A) ϕ(ab) = ϕ(ba)} whenever ϕ∈ A∗,
G = {g ∈ A|g invertible},
A
U = {u ∈ A| uu∗ = u∗u= 1} ⊆ G ,
A A
Asa = {a ∈ A| a= a∗},
(A∗)sa = {ϕ ∈A∗ | (∀a∈ A) ϕ(a∗)= ϕ(a)}.
In the special case of a W∗-algebra M we will also use the notation
M ={ϕ ∈ M∗ | ϕ is w∗-continuous},
∗
Msa =M ∩(M∗)sa.
∗ ∗
Proposition 2.2 Let A be a unital C∗-algebra having a faithful tracial state τ: A → C. Con-
sider the mapping
Θτ: A → A∗, a7→ Θτ,
a
Belti¸t˘a and Ratiu: Geometric representations 3
where for each a∈ A we define
Θτ: A → C, Θτ(b) := τ(ab).
a a
The mapping Θτ has the following properties:
(a) KerΘτ = {0}.
(b) AΘτa = {a}′ for all a∈ A.
(c) The mapping Θτ is G -equivariant with respect to the adjoint action of G on A and the
A A
coadjoint action of G on A∗. In particular, the mapping
A
Θτ| : Asa → (A∗)sa
Asa
is U -equivariant with respect to the adjoint action of U on Asa and the coadjoint action
A A
of U on (A∗)sa.
A
(d) For each a ∈ A the mapping Θτ induces a bijection of the adjoint orbit G ·a onto the
A
coadjoint orbit G ·Θτ. In particular, if a∈ Asa and there exists a conditional expectation
A a
of A onto {a}′ then we have a commutative diagram of U -equivariant diffeomorphisms
A
of Banach manifolds
U ·a
A
ցΘτ
x
UA/U{a}′ −−−−→ UA ·Θτa
(e) If, moreover, AisaW∗-algebraandthefaithfultracial stateτ isnormal, thenRanΘτ ⊆ A
∗
and the hypothesis on conditional expectation from (d) holds for each a∈ Asa.
Proof. This is proved like Proposition 2.12 in [10]. (cid:4)
Corollary 2.3 Let A be a unital C∗-algebra having a faithful tracial state τ: A → C and let
a = a∗ ∈ A be such that there exists a conditional expectation of A onto {a}′. Then the unitary
orbit of a has a natural structure of U -homogeneous weakly symplectic manifold.
A
Proof. Use Proposition 2.2 along with the reasoning that leads to Corollary 2.9 in [32]. (cid:4)
Corollary 2.4 InaW∗-algebraM thatadmits afaithfulnormal tracial state the unitaryorbitof
eachself-adjointelementhasanaturalstructure ofU -homogeneous weaklysymplectic manifold.
M
Proof. Use Proposition 2.2 along with Corollary 2.9 in [10]. (cid:4)
Proposition 2.5 Let A be a unital C∗-algebra and a = a∗ ∈ A have the spectrum a finite
set. Then the unitary orbit of a has a natural structure of U -homogeneous complex Banach
A
manifold.
Belti¸t˘a and Ratiu: Geometric representations 4
Proof. Let a = a∗ ∈ A such that there exist finitely many different numbers λ ,...λ ∈ R
1 m
with
a = λ e +···+λ e ,
1 1 m m
where e ,...,e ∈ A are orthogonal projections satisfying e e = 0 whenever i 6= j and e +
1 m i j 1
···+e = 1. It is clear that, denoting p = e +···+e for j = 1,...,m−1, we have
m j 1 j
{a}′ = {b ∈ A| be = e b for 1 ≤ j ≤ m}= {b ∈ A| bp = p b for 1 ≤ j ≤ m−1}.
j j j j
Thus both the unitary orbit of a and the unitary orbit of (p ,...,p ) ∈ A × ··· × A can
1 m−1
be identified with UA/U{a}′, and now the desired conclusion follows by Corollary 16 in [9] (see
Proposition 2.7 below for the special case of W∗-algebras). (cid:4)
Theorem 2.6 If a unital C∗-algebra A possesses a faithful tracial state, then the unitary orbit
of each self-adjoint element with finite spectrum has a natural structure of an U -homogeneous
A
weakly Ka¨hler manifold.
Proof. The proof is similar to the one of Proposition 4.8 in [10]. (cid:4)
We conclude this section by a result that will be needed in the proof of Theorem 5.8. We
say that a unital W∗-algebra M is finite if for every u ∈ M with u∗u = 1 we have uu∗ = 1 as
well. In this case each projection p ∈ M is finite, in the sense that, if v ∈ M, v∗v = p and vv∗
is a projection smaller than p, then vv∗ = p. Clearly a unital W∗-algebra is finite if it has a
faithful tracial state.
Proposition 2.7 Let M be a finite W∗-algebra and e ,...,e ∈ M orthogonal projections sat-
1 n
isfying e e = 0 whenever i 6= j and e + ··· + e = 1. Next consider the sub-W∗-algebra
i j 1 n
M = {a ∈ M | ae = e a for j = 1,...,m} of M, the subgroup P = {g ∈ G | e ge = 0 if 1≤
0 j j M k j
j < k ≤ n} of G , and define the mapping
M
ψ: U /U → G /P, uU 7→ uP.
M M0 M M0
Then ψ is a real analytic U -equivariant diffeomorphism.
M
Proof. The mapping ψ is clearly real analytic. Next note that U ∩P = U , hence ψ is
M M0
injective.
The fact that ψ is surjective can be equivalently expressed by the following assertion: For
all g ∈ G there exist u ∈ U and q ∈ P such that g = uq. In order to prove this fact, denote
M M
p = 0 and p = e +···+e for j = 1,...,n, and note that
0 j 1 j
P = {g ∈ G |gp = p gp for j = 1,...,p }.
M j j j n
We have
p ≤ p ≤ ··· ≤ p and l(gp )≤ l(gp )≤ ··· ≤l(gp ).
1 2 n 1 2 n
Now let us fix j ∈ {1,...,n}. Recall that for each b ∈ M one denotes by l(b) the smallest
projection p ∈ M satisfying pb = b. By Lemma 3.2(iii) in [8] we have l(gp ) ∼ p and l(gp )∼
j j j−1
p . Since p is a finite projection, it then follows that l(gp )−l(gp ) ∼ p −p = e (see
j−1 j j j−1 j j−1 j
for instance Exercise 6.9.8 in [26]). In other words, there exists v ∈M such that v v∗ = e and
j j j j
v∗v = l(gp )−l(gp ).
j j j j−1
Belti¸t˘a and Ratiu: Geometric representations 5
Now, denoting
u= v +···+v ∈ M
1 n
it is easy to check that u∈ U and u∗e u= l(gp )−l(gp ) for j =1,...,n. By summing up
M j j j−1
these equalities for j = 1,...,k, we get u−1l(gp ) =p u∗ for each k ∈ {1,...,n}.
k k
It then follows that for every k we have l(u−1l(gp )) = l(p u∗), whence l(u−1gp ) = p by
k k k k
Lemma 3.2(i) in [8]. Consequently q := u−1g ∈ P, and we are done. (cid:4)
3 Homogeneous vector bundles
The present section is devoted to a general construction of homogeneous vector bundles in a
C∗-algebraic setting.
Construction 3.1 LetAbeaunitalC∗-algebra, B aunitalsub-C∗-algebraofAandϕ: A→ C
a state such that there exists a conditional expectation E: A → B with ϕ ◦ E = ϕ. By
conditional expectation we mean that E is a boundedlinear mapping such that kEk = 1 and
E is idempotent, that is, E2 = E. It then follows by the theorem of Tomiyama (see [37] or [34])
that E has the additional properties:
E(a∗)= E(a)∗, (3.1)
0≤ E(a)∗E(a) ≤ E(a∗a), (3.2)
E(b ab )= b E(a)b , (3.3)
1 2 1 2
for all a∈ A and b ,b ∈ B.
1 2
Now let
ρ: A→ B(H) and ρ : B → B(H )
ϕ ϕ
be the GNS unital ∗-representations of A and B corresponding to ϕ and ϕ| , respectively. We
B
recall that, for instance, H is the Hilbert space obtained from A by factoring out the null-space
of the nonnegative definite Hermitian sesquilinear form
A×A∋ (a ,a )7→ ϕ(a∗a ) ∈ C
1 2 2 1
and then taking the completion, and for each a ∈ A the operator ρ(a): H → H is the one
obtained from the linear mapping a′ 7→ aa′ on A. The Hilbert space H is obtained similarly,
ϕ
using the restriction of the aforementioned sesquilinear form to B. It is easy to see that H is
ϕ
a closed subspace of H and for all b ∈ B we have a commutative diagram
ρ(b)
A → H −−−−→ H
E PHϕ PHϕ (3.4)
  
By→ Hy −−ρϕ−(−b→) Hy
ϕ ϕ
where P : H → H denotes the orthogonal projection, while A → H and B → H are the
Hϕ ϕ ϕ
inclusions. The fact that E: A → B is continuous with respect to the GNS scalar products on
A and B (whence its “extension” by continuity to P ) follows since for all a ∈ A we have by
Hϕ
inequality (3.2) that E(a)∗E(a) ≤E(a∗a), hence ϕ(E(a)∗E(a)) ≤ ϕ(E(a∗a)) = ϕ(a∗a).
Belti¸t˘a and Ratiu: Geometric representations 6
By restriction we get a norm-continuous unitary representation of the unitary group of B,
ρ| : U → U .
UB B B(Hϕ)
Since U is a Lie group with the topology inherited from U and the self-adjoint mapping E
B A
gives a continuous projection of the Lie algebra of U onto the Lie algebra of U , it then follows
A B
by Proposition 11 in Chapter III, §1.6 in [13] that we have a principal U -bundle
B
π : U → U /U , u 7→ u·U .
B A A B B
Now we can construct the U -homogeneous vector bundle associated with π and ρ| , which
A B UB
we denote by
Π : D → U /U
ϕ,B ϕ,B A B
(see for instance 6.5.1 in [12] for this construction). We recall that D = U × H , in the
ϕ,B A UB ϕ
sense that D is the quotient of U ×H by the equivalence relation ∼ defined by
ϕ,B A ϕ
(u ,f )∼ (u ,f ) ⇐⇒ (∃v ∈ U ) u = u v and f = ρ (v−1)f ,
1 1 2 2 B 1 2 1 ϕ 2
while the mapping Π takes the equivalence class [(u,f)] of any pair (u,f) to u·U .
ϕ,B B
Definition 3.2 With the notations of Construction 3.1, the U -homogeneous vector bundle
A
Π : D → U /U is called the homogeneous bundle associated with ϕ and B.
ϕ,B ϕ,B A B
Remark 3.3 In the setting of Construction 3.1, there exist additional structures on the vector
bundle Π : D → U /U , which we discuss below.
ϕ,B ϕ,B A B
(i) Denote by α : U ×U /U → U /U the natural action of U on U /U . Define the
B A A B A B A A B
mapping
β : U ×D → D
ϕ,B A ϕ,B ϕ,B
that takes the pair consisting of u′ ∈ U and the equivalence class of (u,f) ∈U ×H to
A A ϕ
the equivalence class of (u′u,f). Then β is a real analytic action of U on D and
ϕ,B A ϕ,B
the diagram
U ×D −−β−ϕ,−B→ D
A ϕ,B ϕ,B
idUA×Πϕ,B Πϕ,B
 
UA ×UyA/UB −−α−B−→ UAy/UB
is commutative.
(ii) The bundle Π is actually a Hermitian vector bundle, in the sense that its fibers come
ϕ,B
equipped with structures of complex Hilbert spaces, and moreover for each u ∈ U the
A
mapping β (u,·): D → D is (bounded linear and) fiberwise unitary.
ϕ,B ϕ,B ϕ,B
Example 3.4 Let A be a unital C∗-algebra and ϕ: A → C a state with the corresponding
GNS representation ρ: A → B(H). We describe below the extreme situations of the unital
sub-C∗-algebra B of A, when the hypothesis of Construction 3.1 is satisfied.
(i) For B = A we can take E = id . In this case the homogeneous vector bundle associated
A
with ϕ and B is just the vector bundle with the base reduced to a single point and with
the fiber equal to H.
Belti¸t˘a and Ratiu: Geometric representations 7
(ii) TheotherextremesituationisforB = C1,whenwecantakeE(·) = ϕ(·)1. Then,denoting
T = {z ∈ C | |z| = 1}, it follows that the homogeneous vector bundle associated with ϕ
and B is a line bundle whose base is the Banach-Lie group U /T1.
A
Example 3.5 If M is a W∗-algebra and 0 ≤ ϕ ∈ M , then there always exists a conditional
∗
expectation
E : M → Mϕ with ϕ◦E = ϕ.
ϕ ϕ
We recall that the existence of E follows by Theorem 4.2 in Chapter IX of [36] in the case
ϕ
when ϕ is faithful. In the general case, denote by p = p∗ = p2 ∈ M the support of ϕ. Since in a
W∗-algebra every element is a linear combination of unitary elements, it follows by Lemma 2.7
in [10] that, denoting ϕ := ϕ| ∈ (pMp) , we have
p (pMp) ∗
Mϕ = {a∈ M | ap = pa and pap ∈(pMp)ϕp} = (pMp)ϕp ⊕(1−p)M(1−p).
Now, since ϕ is faithful on pMp, it follows by the aforementioned theorem from [36] that there
p
exists a conditional expectation E : pMp → (pMp)ϕp with ϕ ◦E =E . Then
ϕp p ϕp ϕp
E : M → Mϕ, a 7→ E (pap)+(1−p)a(1−p)
ϕ ϕp
is a conditional expectation with ϕ◦E = ϕ (see also Remark 2.5 in [10]).
ϕ
Example 3.6 Let A be a unital C∗-algebra, B a unital sub-C∗-algebra of A and E: A → B
a conditional expectation as in Construction 3.1. If ϕ : B → C is a pure state of B, then it
0
is easy to see that ϕ := ϕ ◦E: A → C is in turn a pure state of A provided it is the unique
0
extension of ϕ to A. Since pure states lead to irreducible representations, this easy remark can
0
be viewed as a version of Theorem 2.5 in [11] asserting that (roughly speaking) irreducibility of
the isotropy representation implies global irreducibility. (Compare also Theorem 5.4 below.)
We refer to the papers [4] and [5] for situations when, converse to the above remark, the
existence of a unique conditional expectation from A onto B follows from the assumption that
each pure state of B extends uniquely to a pure state of A.
4 Reproducing kernels
In what follows, by continuous vector bundle we mean a continuous mapping Π: D → T,
where D and T are topological spaces, the fibers D := Π−1(t) (t ∈ T) are complex Banach
t
spaces, and Π is locally trivial with fiberwise bounded linear trivializations. We say that Π is
Hermitian if it is equipped with a continuous Hermitian structure
(· |·): D⊕D → C
thatmakeseachfiberD intoacomplexHilbertspace. Wedefineinasimilarmannerthesmooth
t
vector bundles and holomorphic vector bundles. For instance, for a Hermitian holomorphic
vector bundle we require that both D and T are complex Banach manifolds, Π is holomorphic
andtheHermitianstructureissmooth. WedenotebyC(T,E), C∞(T,E)andO(T,E)thespaces
of continuous, C∞ and holomorphic sections of the bundle Π: T → E, respectively, whenever
the corresponding smoothness condition makes sense. We refer to [27] and [1] for basic facts on
vector bundles.
Belti¸t˘a and Ratiu: Geometric representations 8
Definition 4.1 Let Π: D → T be a continuous Hermitian vector bundle, p ,p : T ×T → T
1 2
the natural projections, and
p∗Π: p∗D → T ×T
j j
the pull-back of Π by p , j = 1,2, which are in turn Hermitian vector bundles. Also consider
j
the continuous vector bundle Hom(p∗Π,p∗Π) over T ×T, whose fiber over (s,t)∈ T ×T is the
2 1
Banach space B(D ,D ).
t s
A positive definite reproducing kernel on Π is a continuous section
K ∈ C(T ×T,Hom(p∗Π,p∗Π))
2 1
having the property that for every integer n ≥ 1 and all choices of t ,...,t ∈ T and ξ ∈
1 n 1
D ,...,ξ ∈ D , we have
t1 n tn
n
(K(t ,t )ξ | ξ ) ≥ 0.
l j j l Dtl
jX,l=1
If Π is a holomorphic vector bundle, we say that the reproducing kernel K is holomorphic if
for each t ∈ T and every ξ ∈ D we have K(·,t)ξ ∈ O(T,D).
t
Thefollowing properties of a reproducingkernel K are immediate consequences of the above
definition:
(i) For all t ∈ T we have K(t,t)≥ 0 in B(D ).
t
(ii) For all t,s ∈ T we have K(t,s)∈ B(D ,D ), K(s,t) ∈B(D ,D ) and K(t,s)∗ = K(s,t).
s t t s
Thefirstpart of the next statement is a version of Theorem 1.4 in [11]. Theproof consists in
adapting to the present setting a number of basic ideas from the classical theory of reproducing
kernel Hilbert spaces (see for instance [25] and [19]).
Theorem 4.2 Let Π: D → T be a continuous Hermitian vector bundle, denote by p ,p : T ×
1 2
T → T the projections and consider K ∈ C(T ×T,Hom(p∗Π,p∗Π)). Then K is a reproducing
2 1
kernel on Π if and only if there exists a linear mapping ι: H → C(T,D), where H is a complex
Hilbert space and evι := (ι(·))(t): H → D is a bounded linear operator for all t ∈T, such that
t t
K(s,t)= evι(evι)∗: D → D , (4.1)
s t t s
for all t ∈ T. If this is the case, then the mapping ι can be chosen to be injective. If, moreover,
Π is a holomorphic vector bundle then the reproducing kernel K is holomorphic if and only if
any mapping ι as above has the range contained in O(T,D).
Proof. Given a mapping ι as described in the statement, it is clear that K defined by (4.1)
is a reproducing kernel.
Conversely, given the reproducing kernel K, consider the complex linear subspace HK ⊂
0
C(T,D) spanned by {K := K(·,Π(ξ))ξ | ξ ∈ D} and denote by HK its completion with respect
ξ
to the scalar product defined by
n
(Θ |∆) = (K(Π(ξ ),Π(η ))η |ξ ) ,
HK l j j l DΠ(ξl)
jX,l=1
Belti¸t˘a and Ratiu: Geometric representations 9
n n
where Θ = K(·,Π(η ))η , ∆ = K(·,Π(ξ ))ξ ∈ HK. In order to see that this formula
j j l l 0
jP=1 lP=1
is independent on the choices used to define Θ and ∆, let us first note the reproducing kernel
property
(Θ |K ) = (Θ(Π(ξ)) |ξ) (4.2)
ξ HK DΠ(ξ)
which holds whenever ξ ∈ D and Θ ∈HK. Now, for Θ and ∆ as above we get
0
n
(Θ |∆) = (Θ(Π(ξ )) |ξ ) .
HK l l DΠ(ξl)
Xl=1
Hence if Θ = 0, then (Θ | ∆) = 0.
HK
Moreover,
n
(Θ |Θ) = (K(Π(η ),Π(η ))η | η ) ≥ 0,
HK l j j l DΠ(ηl)
jX,l=1
which shows that (·| ·) is a nonnegative definite Hermitian sesquilinear form on HK. There-
HK 0
fore, it satisfies the Cauchy-Schwarz inequality and, in particular, the reproducing kernel prop-
erty (4.2) implies that
1/2 1/2
|(Θ(Π(ξ)) | ξ) |≤ (Θ |Θ) (K |K ) for all ξ ∈ D.
DΠ(ξ) HK ξ ξ HK
Thus, if (Θ |Θ) = 0, then Θ= 0. On the other hand, for each ξ ∈ D we have
HK
(K | K ) = (K(Π(ξ),Π(ξ))ξ | ξ) ≤ kK(Π(ξ),Π(ξ))kkξk2 .
ξ ξ HK DΠ(ξ) DΠ(ξ)
Now it follows by this inequality along with the previous one that for all Θ ∈ HK and t ∈T we
0
have
kΘ(t)k ≤ kK(t,t)k1/2(Θ |Θ)1/2 = kK(t,t)k1/2kΘk .
Dt HK HK
Consequently we can uniquely define for each t ∈T the bounded linear mapping evι: HK → D
t t
satisfying evι(Θ) = Θ(t) for all Θ ∈ HK. Then we define ι: HK → C(T,D) by (ι(·))(t) :=
t 0
evι(·): HK → D for all t ∈ T. Now note that the reproducing kernel property (4.2) extends by
t t
continuity to arbitrary Θ∈ HK under the following form:
(Θ |K ) = ((ι(Θ))(Π(ξ)) | ξ) for all Θ ∈ HK and ξ ∈ D. (4.3)
ξ HK DΠ(ξ)
In particular, (4.3) shows that ι: HK → C(T,D) is injective: if ι(Θ) = 0, then Θ ⊥ K in HK
ξ
for all ξ ∈ D, hence Θ = 0 since the linear span of {K | ξ ∈ D} is dense in HK. The proof of
ξ
the first part of the statement is finished.
Now assume that Π is a holomorphic vector bundle and let ι be a corresponding injective
mapping. Note that the expression (4.1) of K in terms of the mappings evι implies that for
s
each t ∈ T and ξ ∈ D we have
t
K(s,t)ξ = evι((evι)∗ξ) = ι((evι)∗ξ)(s) for all s ∈ T. (4.4)
s t t
Thus, if Ranι⊆ O(T,D) then K is a holomorphic kernel. Conversely, assume that ι is injective
and the kernel K is holomorphic. Then (4.4) shows that ι(h) ∈ O(T,D) for all h ∈ H :=
0
Ran(evι )∗ (⊆ H). Note that in H we have
Π(ξ)
ξS∈D
H⊥ = (Ran(evι )∗)⊥ = Ker(evι )= {0},
0 Π(ξ) Π(ξ)
ξ\∈D ξ\∈D
Belti¸t˘a and Ratiu: Geometric representations 10
where the latter equality follows since ι is injective. Consequently we know that ι(h) ∈O(D,T)
when h runs through a dense linear subspace of H. Now note that, as above, we can show that
k(ι(h))(t)k ≤kK(t,t)kkhk for all h ∈ H,
Dt H
hence ι: H → C(D,T) is continuous when C(D,T) is equipped with the topology of uniform
convergence on the compact subsets of D. On the other hand, O(D,T) is a closed subspace of
C(D,T) with respect to that topology (see e.g., Corollary 1.14 in [38])). Since we have already
seen that ι(H ) ⊆ O(D,T) and H is dense in H, it then follows that Ranι ⊆ O(D,T), as
0 0
desired. (cid:4)
5 Reproducing kernels and GNS representations
In this section we obtain our main results concerning geometric realizations of GNS represen-
tations in spaces of sections of Hermitian vector bundles (see Theorems 5.4 and 5.8). Since the
corresponding spaces of sections will be reproducing kernel Hilbert spaces, we begin by con-
structing the reproducing kernels. In the extreme case B = C1 (see Example 3.4(ii)) the kernel
K from the following construction is related to a certain hermitian kernel studied in [18]
ϕ,B
(more precisely, see formula (6.2) in [18]).
Construction 5.1 LetAbeaunitalC∗-algebra, B aunitalsub-C∗-algebraofAandϕ: A→ C
a state such that there exists a conditional expectation E: A → B with ϕ◦E = ϕ. Recall from
Definition 3.2 the homogeneous vector bundle
Π : D = U × H → U /U
ϕ,B ϕ,B A UB ϕ A B
associated with ϕ and B. Then we define
ι : H → C(U /U ,D ), by ι (h)(uU ) = [(u,P (ρ(u)−1h))]
ϕ,B A B ϕ,B ϕ,B B Hϕ
for all h ∈H and u∈ U . We also define
A
K ∈ C U /U ×U /U ,Hom(p∗(Π ),p∗(Π ))
ϕ,B A B A B 2 ϕ,B 1 ϕ,B
(cid:0) (cid:1)
by
K (u U ,u U )[(u ,f)]= [(u ,P (ρ(u−1u )f))],
ϕ,B 1 B 2 B 2 1 Hϕ 1 2
where u ,u ∈ U , f ∈ H , and p ,p : U /U ×U /U → U /U are the projections.
1 2 A ϕ 1 2 A B A B A B
Wenotethatbothι andK arewelldefinedbythecommutativity ofthediagram (3.4).
ϕ,B ϕ,B
Definition 5.2 In the setting of Construction 5.1, the operator ι : H → C(U /U ,D ) is
ϕ,B A B ϕ,B
called the realization operator associated with ϕ and B and the map K ∈ C U /U ×
ϕ,B A B
U /U ,Hom(p∗(Π ),p∗(Π )) is called the reproducing kernel associated with(cid:0) ϕ and B.
A B 2 ϕ,B 1 ϕ,B
(cid:1)
Remark 5.3 (i) It will follow from the proof of Theorem 5.4 below that the reproducing
kernel K associated with a state ϕ and B is indeed a reproducing kernel in the sense
ϕ,B
of Definition 4.1.