A complex semigroup approach to group algebras of infinite dimensional Lie groups
aa r X i v : . [ m a t h . OA ] S e p A complex semigroup approach to groupalgebras of infinite dimensional Lie groups
Karl-Hermann NeebNovember 2, 2018
To K. H. Hofmann on the occasion of his 75th birthday
Abstract
A host algebra of a topological group G is a C ∗ -algebra whose rep-resentations are in one-to-one correspondence with certain continuousunitary representations of G . In this paper we present an approachto host algebras for infinite dimensional Lie groups which is based oncomplex involutive semigroups. Any locally bounded absolute value α on such a semigroup S leads in a natural way to a C ∗ -algebra C ∗ ( S, α ),and we describe a setting which permits us to conclude that this C ∗ -algebra is a host algebra for a Lie group G . We further explain howto attach to any such host algebra an invariant weak- ∗ -closed convexset in the dual of the Lie algebra of G enjoying certain nice convexgeometric properties. If G is the additive group of a locally convexspace, we describe all host algebras arising this way. The general non-commutative case is left for the future.Keywords: complex semigroup, infinite dimensional Lie group, hostalgebra, multiplier algebra, unitary representation.MSC: 22E65, 22E45. Introduction If G is a locally compact group, then Haar measure on G leads to the convo-lution algebra L ( G ), and we obtain a C ∗ -algebra C ∗ ( G ) as the enveloping C ∗ -algebra of L ( G ). This C ∗ -algebra has the universal property that each1continuous) unitary representation ( π, H ) of G on some Hilbert space H de-fines a unique non-degenerate representation of C ∗ ( G ) on H and, conversely,each non-degenerate representation of C ∗ ( G ) arises from a unique unitaryrepresentation of G . This correspondence is a central tool in the harmonicanalysis on G because the well-developed theory of C ∗ -algebras provides apowerful machinery to study the set of all irreducible representations of G ,to endow it with a natural topology and to understand how to decomposerepresentations into irreducibles or factor representations.For infinite dimensional Lie groups, i.e., Lie groups modeled on infinite di-mensional locally convex spaces, there is no natural analog of the convolutionalgebra L ( G ), so that we cannot hope to find a C ∗ -algebra whose represen-tations are in one-to-one correspondence to all unitary representations of G .However, in [Gr05] H. Grundling introduces the notion of a host algebra ofa topological group G . This is a pair ( A , η ), consisting of a C ∗ -algebra A and a morphism η : G → U ( M ( A )) of G into the unitary group of its mul-tiplier algebra M ( A ) with the following property: For each non-degeneraterepresentation π of A and its canonical extension e π to M ( A ), the unitaryrepresentation e π ◦ η of G is continuous and determines π uniquely. In thissense, A is hosting a certain class of representations of G . A host algebra A is called full if it is hosting all continuous unitary representations of G .Now it is natural to ask to which extent infinite dimensional Lie groups, orother non-locally compact groups, possess host algebras. One cannot expectthe existence of a full host algebra because, f.i., the topological dual E ′ of aninfinite dimensional locally convex space E carries no natural locally com-pact topology. Therefore one is looking for host algebras that accommodatecertain classes of continuous unitary representations.In the present paper we discuss a construction of host algebras basedon holomorphic extensions of unitary representations of a Lie group G tocertain complex semigroups S . Some of the basic ideas of our constructionsappear already in [Ne95], where one finds the construction of the enveloping C ∗ -algebras C ∗ ( S, α ) of a complex involutive semigroup S , endowed with alocally bounded absolute value α , and also in [Ne98], where this is appliedto the special case where S is a complex Banach–Lie group. Here we addressthe situation where S may be an infinite dimensional semigroup which is nota group.The structure of the paper is as follows. In Section 1 we first recall theconcept of a complex involutive semigroup S and associate to any locallybounded absolute value α on S a C ∗ -algebra C ∗ ( S, α ) with a holomorphic2orphism η α : S → C ∗ ( S, α ) having a suitable universal property. Sinceour goal is to construct host algebras for infinite dimensional Lie groups, webuild in Section 2 a bridge between complex involutive semigroups and Liegroups by defining the notion of a host semigroup of a Lie group. Roughlyspeaking, this a complex involutive semigroup S on which the Lie group G acts smoothly by unitary multipliers and for which there exists an openconvex cone W in the Lie algebra L( G ) of G , invariant under the adjointaction, for which all R -actions on S defined by the one-parameter semi-groups γ x ( t ) = exp G ( tx ), x ∈ W , extend to “holomorphic” one-parametersemigroups C + = R + i [0 , ∞ [ → M ( S ) mapping the open upper halfplane C holomorphically into S ⊆ M ( S ). The main result of Section 2 is that foreach locally bounded absolute value α on a host semigroup S , the C ∗ -algebra C ∗ ( S, α ) is a host algebra of G .This leaves us with the problem to understand the classes of represen-tations of G hosted by such C ∗ -algebras. To clarify this point, we considerin Section 3 multiplier actions η : G → U ( M ( A )) of a Lie group G on a C ∗ -algebra A and study to which extent the action of certain one-parametersemigroups of G extends holomorphically to the upper halfplane. This leadsto the momentum mapΨ η : S ( A ) ∞ → L( G ) ′ , ϕ i d ( e ϕ ◦ η )( ) . Here S ( A ) ∞ denotes the set of all states ϕ of A for which the canonicalextension e ϕ to M ( A ) yields a smooth function e ϕ ◦ η : G → C . The weak- ∗ -closed hull I η of the image of Ψ η is a convex set invariant under the coadjointaction, called the momentum set of ( A , η ). A crucial observation is that, ifthe multiplier action comes from a host algebra C ∗ ( S, α ), where S is a hostsemigroup, the convex cone B ( I η ) := { x ∈ L( G ) : inf h I η , x i > −∞} has non-empty interior and the support function s : B ( I η ) → R , x
7→ − inf h I η , x i is locally bounded. This observation suggests that to find host algebras for G , one should start with an Ad ∗ ( G )-invariant weak- ∗ -closed convex subset C ⊆ L( G ) ′ for which the corresponding support function s C : B ( C ) → R islocally bounded. As the function s C is convex, we take in Section 4 a closer3ook at convex functions on open convex domains in locally convex spaces. Inparticular, we show that whenever L( G ) is barrelled, the existence of interiorpoints in the cone B ( C ) automatically implies that s C is locally boundedand even continuous.In Section 5 we then show how this circle of ideas can be completed inthe abelian case. Here the Lie group G is a locally convex space V andthe semigroup is a tube S = V + iW , where W ⊆ V is an open con-vex cone. In this case it suffices to consider absolute values of the form α ( a + ib ) = e − inf h C,b i , where C ⊆ V ′ is a weak- ∗ -closed convex subset. Now α is locally bounded if and only if the support function s C ( x ) = − inf h C, x i is locally bounded on W . If this is the case, then the results of Section 4imply that C is locally compact and C ∗ ( S, α ) ∼ = C ( C ) is a host algebraof V hosting precisely all unitary representations of V arising from spectralmeasures on the locally compact subset C ⊆ V ′ .Section 6 contains a brief discussion of the finite dimensional case, whichis developed in detail in [Ne99]. Here we give a short and direct proof of thefact that any host algebra of G coming from a host semigroup is a quotientof the group C ∗ -algebra C ∗ ( G ).The next steps of this project aim at a better understanding of theclasses of representations of a Lie group G hosted by C ∗ -algebras of theform C ∗ ( S, α ). The first major problem one has to solve here is to find asuitable complex involutive semigroup S whenever the invariant convex set C ⊆ L( G ) ′ is given. For finite dimensional groups this has been carried out in[Ne99], but for infinite dimensional groups many key tools are still missing.Furthermore, once the semigroup S is constructed, one has to find the classof unitary representations of G extending to holomorphic representations of S . We leave all that to the future. Preliminaries
For the sake of easier reference, we collect some of the basic definitions con-cerning infinite dimensional manifolds and Lie groups.Let X and Y be locally convex topological vector spaces, U ⊆ X openand f : U → Y a map. Then the derivative of f at x in the direction of h isdefined as df ( x )( h ) := lim t → t (cid:0) f ( x + th ) − f ( x ) (cid:1) whenever the limit exists. The function f is called differentiable at x if4 f ( x )( h ) exists for all h ∈ X . It is called continuously differentiable or C ifit is continuous and differentiable at all points of U and df : U × X → Y, ( x, h ) df ( x )( h )is a continuous map. It is called a C n -map if it is C and df is a C n − -map, and C ∞ (or smooth ) if it is C n for all n ∈ N . This is the notion ofdifferentiability used in [Mil84], [Ha82] and [Gl02], where the latter referencedeals with the modifications needed for incomplete spaces. If X and Y arecomplex, f is called holomorphic if it is smooth and its differentials df ( x ) arecomplex linear. If Y is Mackey complete, it suffices that f is C .Since we have a chain rule for C -maps between locally convex spaces,we can define smooth manifolds as in the finite dimensional case. A chart( ϕ, U ) with respect to a given manifold structure on M is an open set U ⊂ M together with a homeomorphism ϕ onto an open set of the model space.A Lie group G is a group equipped with a smooth manifold structuremodeled on a locally convex space for which the group multiplication andthe inversion are smooth maps. We write ∈ G for the identity element and λ g ( x ) = gx , resp., ρ g ( x ) = xg for the left, resp., right multiplication on G .Then each x ∈ T ( G ) corresponds to a unique left invariant vector field x l with x l ( ) = x . The space of left invariant vector fields is closed under theLie bracket, hence inherits a Lie algebra structure. We thus obtain on thetangent space T ( G ) a continuous Lie bracket which is uniquely determinedby [ x, y ] l = [ x l , y l ] for x, y ∈ T ( G ). We write L( G ) := ( T ( G ) , [ · , · ]) for the so-obtained topological Lie algebra. Then L defines a functor from the categoryof locally convex Lie groups to the category of locally convex topological Liealgebras. The adjoint action of G on L( G ) is defined by Ad( g ) := L( c g ),where c g ( x ) = gxg − . This action is smooth and each Ad( g ) is a topologicalisomorphism of L( G ). The coadjoint action on the topological dual space L( G ) ′ is defined by Ad ∗ ( g ) .f := f ◦ Ad( g ) − and all these maps are continuouswith respect to the weak- ∗ -topology on L( G ) ′ , but in general the coadjointaction of G is not continuous with respect to this topology. C ∗ -algebras associated to complex semigroups In this section we associate to each complex involutive semigroup S , endowedwith an absolute value α , a C ∗ -algebra C ∗ ( S, α ). As we shall see later on, one5an use these semigroup algebras to construct host algebras for Lie groups,and this is our main main purpose for their construction.
Definition 1.1 (a) An involutive complex semigroup is a complex manifold S modeled on a locally convex space which is endowed with a holomorphicsemigroup multiplication and an antiholomorphic antiautomorphism denoted s s ∗ .(b) A function α : S → R + is called an absolute value if α ( s ) = α ( s ∗ ) and α ( st ) ≤ α ( s ) α ( t )for all s, t ∈ S .(c) A holomorphic representation ( π, H ) of a complex involutive semi-group S on the Hilbert space H is a morphism π : S → B ( H ) of involu-tive semigroups which is holomorphic if B ( H ) is endowed with its naturalcomplex Banach space structure defined by the operator norm. If α is anabsolute value on S , then the representation π is said to be α -bounded if k π ( s ) k ≤ α ( s ) holds for each s ∈ S . A representation is called non-degenerate if π ( S ) .v = { } implies v = 0. Examples 1.2 (1) If H is a complex Lie group and s s ∗ an antiholomor-phic antiautomorphism H , then H is a complex involutive (semi)group. Anyopen ∗ -subsemigroup of H is a complex involutive semigroup.(2) If V is a locally convex space and W ⊆ V an open convex cone,then S := V + iW ⊆ V C is an involutive subsemigroup with respect to theinvolution ( x + iy ) ∗ := − x + iy .(3) If A is a C ∗ -algebra, then its multiplicative semigroup ( A , · ) is acomplex involutive semigroup and α ( a ) := k a k is an absolute value on A .An important example is the C ∗ -algebra B ( H ) of bounded operators on theHilbert space H .(4) Let A be a unital C ∗ -algebra and τ = τ ∗ ∈ A an involution, i.e., τ = . For a, b ∈ A we write a < b if there exists an invertible element c ∈ A with b − a = c ∗ c . Then S := { s ∈ A : s ∗ τ s < τ } is an open subsemigroup of A with respect to multiplication. To see that itis non-empty, we observe that we may write τ = − p = ( − p ) − p for a6rojection p = p ∗ = p ∈ A . For λ ∈ C × and s := λ ( − p ) + λ − p we thenhave s ∗ τ s = | λ | ( − p ) − | λ − | p < τ = ( − p ) − p if and only if | λ | <
1. The boundary of S contains the real Banach–Lie group U ( A , τ ) := { g ∈ A × : g ∗ τ g = τ } . Definition 1.3
Let S be a complex involutive semigroup and α a locallybounded absolute value on S . We associate to the pair ( S, α ) a C ∗ -algebra C ∗ ( S, α ) as follows.First, we endow the semigroup algebra C [ S ], whose elements we write asfinitely supported functions f : S → C , with the submultiplicative seminorm k f k α := P s ∈ S | f ( s ) | α ( s ) and the involution f ∗ ( s ) := f ( s ). Let ℓ ( S, α ) bethe complex involutive Banach algebra obtained by completion of this semi-normed ∗ -algebra. We define η α ( s ) ∈ ℓ ( S, α ) as the image of the function δ s ( t ) := δ s,t in ℓ ( S, α ) and note that k η α ( s ) k = α ( s ).If A is a C ∗ -algebra, then each homomorphism β : S → ( A , · ) of involutivesemigroups, which is α -bounded in the sense that k β ( s ) k ≤ α ( s ) holds foreach s ∈ S , defines a unique contractive morphism b β : ℓ ( S, α ) → A , b β ( f ) := X s ∈ S f ( s ) β ( s )of Banach- ∗ -algebras satisfying b β ◦ η α = β . Let I E ℓ ( S, α ) denote theintersection of the kernels of all such homomorphism b β for which β is aholomorphic map. On the quotient algebra ℓ ( S, α ) /I , we obtain a C ∗ -normby k [ f ] k := sup β holomorphic k b β ( f ) k ≤ k f k α . We now define C ∗ ( S, α ) as the completion of ℓ ( S, α ) /I with respect to thisnorm. It follows immediately from the construction that we thus obtain a C ∗ -algebra.Before we turn to the universal property of C ∗ ( S, α ), we recall the follow-ing criterion for holomorphy ([Ne99], Cor. A.III.3):
Lemma 1.4
Let M be a complex manifold, V a Banach space and N ⊆ V ′ asubset which is norm-determining, i.e., k v k = sup {| λ ( v ) | : λ ∈ N, k λ k ≤ } for all v ∈ V . Then a locally bounded function f : M → V is holomorphic ifand only if for each λ ∈ N the function λ ◦ f is holomorphic. Theorem 1.5
The C ∗ -algebra C ∗ ( S, α ) has the following properties: (i) There exists a holomorphic morphism η α : S → C ∗ ( S, α ) of involutivesemigroups with total range, i.e., η α ( S ) generates a dense subalgebra. (ii) For each α -bounded holomorphic morphism of involutive semigroups π : S → A to the multiplicative semigroup of a C ∗ -algebra A , there ex-ists a unique morphism of C ∗ -algebras e π : C ∗ ( S, α ) → A with e π ◦ η α = π . Proof. (i) We define η α ( s ) ∈ C ∗ ( S, α ) as the image of the element η α ( s ) ∈ ℓ ( S, α ). Then k η α ( s ) k ≤ k η α ( s ) k = α ( s ) implies that η α is a locally boundedmorphism of involutive semigroups.To see that η α is holomorphic, we first note that the subspace N of con-tinuous linear functionals on C ∗ ( S, α ) spanned by the functionals of the form ϕ ◦ b β on ℓ ( S, α ), where β : S → A is a holomorphic morphism of involutivesemigroups into a C ∗ -algebra A and ϕ ∈ A ′ , separates the points of C ∗ ( S, α )and determines the norm (by definition of the norm on C ∗ ( S, α )). For eachfunctional ψ = ϕ ◦ b β as above the map ψ ◦ η α = ϕ ◦ β : S → C is holomorphic. Therefore Lemma 1.4 implies that η α is holomorphic.That η α ( S ) spans a dense subspace of C ∗ ( S, α ) follows from the construc-tion because the image of S spans a dense subspace of ℓ ( S, α ), hence also inthe quotient by the ideal I .(ii) Let e π : ℓ ( S, α ) → A denote the canonical extension of π which is acontractive morphism of involutive Banach algebras and note that ker e π ⊇ I ,so that e π factors through a morphism ℓ ( S, α ) /I → A of involutive Banachalgebras which, by definition, extends to the completion C ∗ ( S, α ). Remark 1.6 (a) The preceding theorem entails that for each C ∗ -algebra A ,we have Hom( C ∗ ( S, α ) , A ) ∼ = Hom hol (( S, α ) , ( A , k · k ), where the right handside denote the holomorphic contractive morphisms of complex involutivesemigroups with absolute value. This means that C ∗ ( S, α ) defines an adjointof the forgetful functor from the category of C ∗ -algebras to the category8f complex involutive semigroups with absolute value, assigning to a C ∗ -algebra A the semigroup ( A , · , k·k ). It follows in particular that the universalproperty in Theorem 1.5(ii) determines C ∗ ( S, α ) up to isomorphism.(b) If β is an absolute value on S satisfying k η α k ≤ β ≤ α , then thenatural map ϕ : C ∗ ( S, β ) → C ∗ ( S, α ) is an isomorphism with ϕ ◦ η β = η α . Examples 1.7 (a) We take a closer look at the case where S is commuta-tive. Let b S := Hom( S, ( C , · )) \ { } denote the set of non-zero holomorphiccharacters of S , i.e., the one dimensional (=irreducible) non-degenerate rep-resentations. A holomorphic character χ extends to a character of the C ∗ -algebra C ∗ ( S, α ) if and only if it is α -bounded. Hence the set b S α of α -boundednon-zero holomorphic characters form the spectrum of the commutative C ∗ -algebra C ∗ ( S, α ). We conclude that C ∗ ( S, α ) ∼ = C ( b S α ), where b S α ⊆ C ∗ ( S, α ) ′ carries the weak- ∗ -topology. Moreover, the set b S α ∪ { } is weak- ∗ -compact in C ∗ ( S, α ) ′ , and the canonical map η ∗ α : C ∗ ( S, α ) ′ → C S , ϕ ϕ ◦ η is continu-ous with respect to the weak- ∗ -topology on the left and the product topologyon the right. This shows that b S α ∪ { } is compact in C S with respect to theproduct topology which, therefore, coincides with the weak- ∗ -topology de-fined by C ∗ ( S, α ). We conclude that b S α is locally compact with respect tothe product topology and that C ∗ ( S, α ) ∼ = C ( b S α ). We now have k η α ( s ) k = sup {| χ ( s ) | : χ ∈ b S α } , and this absolute value defines the same C ∗ -algebra by Remark 1.6(b).(b) We specialize to the particular case where S = V + iW ⊆ V C holdsfor a real locally convex space V and an open convex cone W ⊆ V . This isan open complex subsemigroup of the complex vector space V C . Any non-zero holomorphic character χ : S → C maps into C × and induces a uniquecontinuous character V → T , hence is of the form χ = e if for some continuouslinear functional f ∈ V ′ (which we also extend to a complex linear functionalon V C ).Now let α be a locally bounded absolute value on S and C α := { f ∈ V ′ : e if ∈ b S α } the set of linear functionals defining α -bounded characters of S . In view of(b), we may w.l.o.g. assume that α ( x + iy ) = k η α ( x + iy ) k = sup { e − f ( y ) : f ∈ C α } = e − inf h C α ,y i C ∗ ( S, α ) or C α . A holomorphic character e if , f ∈ V ′ , is α -bounded, i.e., contained in b S α , if and only if e − f ( y ) ≤ e − inf h C α ,y i for each y ∈ W , which is equivalent to f ( y ) ≥ inf h C α , y i for y ∈ W. (1)This implies that C α is a weak- ∗ -closed convex subset of V ′ , and (a) fur-ther shows that b S α is locally compact as a subset of C S . We continue thediscussion of these examples in Section 5 below.(c) Let S := C = R + i ]0 , ∞ [ ⊆ C be the open upper half plane and α as in (b). Then V = R , W =]0 , ∞ [ and C α ⊆ R is a closed convex subsetbounded from below. Let m := inf C α . Then k η α ( x + iy ) k = e − (inf C α ) · y = e − my , and (1) implies that C α = [ m, ∞ [. Therefore C ∗ ( C , α ) ∼ = C ([ m, ∞ [). Some holomorphic representation theory
Lemma 1.8
Let H be a Hilbert space and π : S → B ( H ) a morphism ofinvolutive semigroups. Then π is a holomorphic representation if and only ifit satisfies the following conditions: (1) π is locally bounded, i.e., for every s ∈ S there exists a neighborhood U such that π ( U ) is a bounded subset of the Banach space B ( H ) . (2) There exists a dense subspace E ⊆ H such that the functions π v ( s ) := h π ( s ) .v, v i are holomorphic for all v ∈ E . Proof.
The necessity of conditions (1) and (2) is obvious, and the conversefollows from [Ne99], Cor. A.III.5, which also holds for general locally convexmanifolds since [He89], Prop. 2.4.9(a) applies to functions on locally convexspaces that are not necessarily complete.
Proposition 1.9
Let S be an involutive complex semigroup and α a locallybounded absolute value on S . (a) If ( π j , H j ) j ∈ J is a set of α -bounded holomorphic representations of S ,then the operators induced by s ∈ S on L j ∈ J H j are bounded, and wethus obtain an α -bounded holomorphic representation of S on the directHilbert space sum cL j ∈ J H . Every non-degenerate α -bounded holomorphic representation is a directsum of cyclic α -bounded holomorphic representations. (c) Every α -bounded cyclic holomorphic representation of S is equivalent toa representation ( π ϕ , H ϕ ) on a reproducing kernel Hilbert space H ϕ ⊆O ( S ) by ( π ( s ) .f )( x ) = f ( xs ) , where the reproducing kernel is given by K ( s, t ) = ϕ ( st ∗ ) for some holomorphic function ϕ ∈ H ϕ . Proof. (a) [Ne99], Prop. IV.2.3; (b) [Ne99], Prop. II.2.11(ii); (c) [Ne99],Lemma IV.2.6.
In this section we describe the connection between Lie groups and complexsemigroups. The key point is that there is a Lie theoretic notion of a hostsemigroup S of a Lie group G which can be used to obtain host algebrasfor G . We start with the definition of a host semigroup of a Lie group andturn in the second subsection to host algebras of topological groups. Definition 2.1
Let S be a complex involutive semigroup. A multiplier of S is a pair ( λ, ρ ) of holomorphic mappings λ, ρ : S → S satisfying the followingconditions: aλ ( b ) = ρ ( a ) b, λ ( ab ) = λ ( a ) b, and ρ ( ab ) = aρ ( b ) . We write M ( S ) for the set of all multipliers of S and turn it into an involutivesemigroup by( λ, ρ )( λ ′ , ρ ′ ) := ( λ ◦ λ ′ , ρ ′ ◦ ρ ) and ( λ, ρ ) ∗ := ( ρ ∗ , λ ∗ ) , where λ ∗ ( a ) := λ ( a ∗ ) ∗ and ρ ∗ ( a ) = ρ ( a ∗ ) ∗ (cf. [FD88], p.778). Remark 2.2
The assignment η S : S → M ( S ) , a ( λ a , ρ a ) defines a mor-phism of involutive semigroups which is surjective if and only if S has anidentity. Its image is an involutive semigroup ideal in M ( S ).11 roposition 2.3 Let α be a locally bounded absolute value on the complexinvolutive semigroup S . Then the following assertions hold: (1) For each non-degenerate α -bounded holomorphic representation ( π, H ) of S there exists a unique unitary representation ( e π, H ) of U ( M ( S )) ,determined by e π ( g ) π ( s ) = π ( gs ) for g ∈ U ( M ( S )) , s ∈ S . (2) There exists a unique homomorphism e η : U ( M ( S )) → U ( M ( C ∗ ( S, α ))) with e η ( g ) η ( s ) = η ( gs ) for g ∈ U ( M ( S )) , s ∈ S . Proof. (a) Every α -bounded holomorphic representation is a direct sumof cyclic ones which in turn are of the form ( π ϕ , H ϕ ) (Proposition 1.9). Wetherefore may assume that π = π ϕ is realized on a reproducing kernel space H ϕ ⊆ O ( S ) with reproducing kernel K ( s, t ) := ϕ ( st ∗ ). Then K is invariantunder the right action of any g = ( λ g , ρ g ) ∈ U ( M ( S )): K ( sg, tg ) = ϕ (( sg )( tg ) ∗ ) = ϕ ( sgg − t ∗ ) = ϕ ( st ∗ ) = K ( s, t ) . Hence e π ϕ ( g )( f ) := f ◦ ρ g defines a unitary operator on H ϕ satisfying e π ϕ ( g ) π ϕ ( s ) = π ϕ ( gs ) for s ∈ S (cf. [Ne99], Remark II.4.5).(b) Since there exists a faithful representation π : C ∗ ( S, α ) → B ( H ), thisfollows directly from (a). Definition 2.4 (a) For a Lie group G with Lie algebra L( G ), we call asmooth function exp G : L( G ) → G an exponential function if for each x ∈ L( G ) the curve γ x ( t ) := exp G ( tx ) is a one-parameter group with γ ′ x (0) = x .In general an exponential function need not exist, but it is unique ([GN07]).(b) A Lie group G with an exponential function is called locally exponen-tial if there exists an open 0-neighborhood U in L( G ) for which exp G | U is adiffeomorphism onto an open subset of G . Definition 2.5
We say that a net ( u i ) i ∈ I in a topological involutive semi-group S is an approximate identity if lim u i s = lim su i = s holds for all s ∈ S . Remark 2.6
For any complex involutive semigroup S with an approximateidentity, the natural map S → M ( S ) is injective. In fact, if ( u i ) is anapproximate identity of S , then the assertion follows from η ( s ) u i = su i → s .We may thus identify S with a subsemigroup of M ( S ).12 efinition 2.7 Let G be a connected Lie group with a smooth exponentialfunction exp G : L( G ) → G . A triple ( S, η, W ), consisting of a complex in-volutive semigroup S with an approximate identity, a group homomorphism η : G → U ( M ( S )) into the group of unitary holomorphic multipliers of S andan open convex Ad( G )-invariant cone W ⊆ L( G ) is called a host semigroupfor G if the following conditions are satisfied:(HS1) The left action of G on S defined by η is smooth.(HS2) For each x ∈ W , the one-parameter group η x : R → U ( M ( S )) , t η (exp G ( tx ))extends to a morphism b η x : C + = R + i [0 , ∞ [ → M ( S )of involutive semigroups defining a continuous left action of the closedupper halfplane C + on S , b η x ( C ) ⊆ S (considered as a subsemigroupof M ( S )), and the corresponding map C → S is holomorphic.(HS3) If f : S → C is a holomorphic function for which all functions f ◦ γ Sx vanish on the open upper half plane, then f = 0. Remark 2.8
Suppose that (
S, η, W ) is a host semigroup of G and x ∈ W .Then the one-parameter subsemigroup b η x ( it ), t >
0, is an approximate iden-tity for S in the sense that for each s ∈ S we havelim t → b η x ( it ) s = lim t → s b η x ( it ) = s because the left action of the closed halfplane C + on S defined by b η x iscontinuous. Proposition 2.9
Let G C be a connected complex locally exponential Lie groupwhose Lie algebra L ( G ) C is the complexification of the real Lie algebra L ( G ) , σ a holomorphic involutive automorphism of G C with L ( σ )( x + iy ) = x − iy ,and G := ( G C ) σ the identity component of the group G σ C of σ -fixed pointsin G C . Let S ⊆ G C be an open connected subsemigroup invariant under theinvolution s ∗ := σ ( s ) − and W ⊆ L ( G ) an open convex invariant cone with exp G C ( iW ) ⊆ S and GSG = S. Then we obtain for each g ∈ G a holomorphic multiplier η ( s ) ∈ U ( M ( S )) by η ( g ) = ( λ g , ρ g ) and ( S, η, W ) is a host semigroup for G . roof. (HS1) follows from the smoothness of the multiplication in G C .(HS2): For x ∈ W we put b η x ( z ) := ( λ exp( zx ) , ρ exp zx ) and note that this isthe multiplier corresponding to the semigroup element exp( zx ) because for z = a + ib we have exp( zx ) = exp( ax ) exp( ibx ) ∈ GS ⊆ S .(HS3): Let f : S → C be a holomorphic function vanishing on the setsexp( C x ), x ∈ W . Let Ω ⊆ exp − ( S ) ⊆ L( G ) C denote the connected compo-nent containing the cone iW . Then the holomorphic function f ◦ exp : Ω → C (cf. [GN07] for the holomorphy of exp) vanishes on iW , hence on a neighbor-hood of iW , and therefore on all of Ω. Since G C is locally exponential, thereexists a point x ∈ iW (sufficiently close to 0) and an open neighborhood U of x in Ω, such that exp( U ) is an open subset of S . Then f vanishes onexp( U ) and hence on all of S because S is connected. Example 2.10
Let G = V be a locally convex space, W ⊆ V an openconvex cone and S := V + iW . Then η ( v )( s ) := s + v yields a host semigroup( S, η, W ) for V . Definition 2.11 If A is a C ∗ -algebra, then we write M ( A ) for the set ofcontinuous linear multipliers on A . Then M ( A ) carries a natural structure ofa C ∗ -algebra and the map η A : A → M ( A ) is injective (cf. [Pe79], Sect. 3.12).We write A for its image in M ( A ). The strict topology on M ( A ) is the locallyconvex topology defined by the seminorms p a ( m ) := k ma k + k am k , a ∈ A , m ∈ M ( A ) . The involution is continuous with respect to this topology and the multipli-cation is continuous on bounded subsets, which implies in particular that theunitary group U ( M ( A )) is a topological group (cf. [Wo95], Sect. 2).For a complex Hilbert space H , we write Rep( A , H ) for the set of non-degenerate representations of A on H . Each representation ( π, H ) of A whichis non-degenerate in the sense that π ( A ) v = { } implies v = 0 extends to aunique representation e π of M ( A ) satisfying e π ◦ η A = π which is continuouswith respect to the strict topology on M ( A ) and the strong operator topologyon B ( H ) (cf. Proposition 8.4 below). Examples 2.12 ([Pe79], Sect. 3.12) (a) If A is a closed ∗ -subalgebra of B ( H ), then M ( A ) ∼ = { X ∈ B ( H ) : X A + A X ⊆ A} A = K ( H ) is the ideal of compact operators in B ( H ), then M ( K ( H )) ∼ = B ( H ).(c) If A = C ( X ) is the C ∗ -algebra of continuous functions vanishing atinfinity on the locally compact space X , then M ( A ) ∼ = C b ( X ) is the C ∗ -algebra of bounded continuous functions on X . Definition 2.13
Let G be a topological group. A host algebra for G isa pair ( A , η ), where A is a C ∗ -algebra and η : G → U ( M ( A )) is a grouphomomorphism such that:(H1) For each non-degenerate representation ( π, H ) of A , the representation e π ◦ η of G is continuous.(H2) For each complex Hilbert space H , the corresponding map η ∗ : Rep( A , H ) → Rep( G, H ) , π e π ◦ η is injective.We say that ( A, η ) is a full host algebra if η ∗ is surjective for each Hilbertspace H . Remark 2.14 (a) If η : G → U ( M ( A )) is strictly continuous , i.e., continu-ous with respect to the strict topology on U ( M ( A )), then Proposition 8.4(3)implies (H1).(b) Since the extension e π of a non-degenerate representation π of A isstrictly continuous, condition (H2) holds if η ( G ) spans a strictly dense sub-algebra of M ( A ). In view of [Wo95], Prop. 2.2, (H2) conversely implies thatspan( η ( G )) is strictly dense in M ( A ).(c) For any multiplier action η : G → U ( M ( A )) of a topological group G on the C ∗ -algebra A , the subspace R , consisting of all elements a ∈ A for which the map G
7→ A , g η ( g ) a is continuous is a right ideal whichis closed because G acts by isometries on A . It is biinvariant under G .Hence A c := R ∩ R ∗ is a C ∗ -subalgebra of A which is G -biinvariant, and thecorresponding homomorphism η c : G → U ( M ( A c )) is strictly continuous.(d) If a homomorphism η : G → U ( M ( A )) satisfies (H2), then (b) impliesthat η ( G ) spans a strictly dense subalgebra of M ( A ). This implies in par-ticular, that the G -biinvariant closed subalgebra A c of A is a two-sided idealof M ( A ) and the corresponding morphism γ : M ( A ) → M ( A c ) is obviouslystrictly continuous and satisfies γ ◦ η = η c .15e claim that ( A c , η c ) also is a host algebra for G . In fact, (H1) followsfrom the strict continuity of η c . Next we note that γ ( M ( A )) contains A c ,so that it is strictly dense in M ( A c ) ([Wo95], Prop. 2.2). Therefore η c ( G ) = γ ( η ( G )) spans a strictly dense subalgebra of M ( A c ), which implies (H2), aswe have seen in (b). Examples 2.15 (a) Let G be a locally compact group and C ∗ ( G ) the en-veloping C ∗ -algebra of the group algebra L ( G ). Then we have a naturalhomomorphism η : G → U ( M ( C ∗ ( G ))) which is determined by the left ac-tion η ( g )( f )( x ) = f ( g − x ) on L -functions. Since G acts continuously fromthe left and the right on L ( G ) and the image of L ( G ) is dense in C ∗ ( G ), η is continuous with respect to the strict topology. It is well known that( C ∗ ( G ) , η ) is a full host algebra of G ([Dix64], Sect. 13.9).(b) Let G be an abelian topological group and b G := Hom( G, T ) its char-acter group. Then any host algebra ( A , η ) for G is commutative because ofthe strict density of η ( G ) in M ( A ). Hence there exists a locally compactspace X with A ∼ = C ( X ) and M ( A ) ∼ = C b ( X ) (cf. Example 2.12). Then U ( M ( A )) ∼ = C ( X, T ), where the strict topology on this group corresponds tothe compact open topology and a ∗ -subalgebra of C b ( X ) is strictly dense ifand only if it separates the points of X ([Br77], Lemma 3.5). Therefore themap γ : X → b G defined by γ ( x )( g ) := η ( g )( x ) is injective, so that we mayconsider X as a subset of the character group b G .If, conversely, X ⊆ b G is a subset, endowed with a locally compact topol-ogy finer than the topology of pointwise convergence on G , then the naturalmap η : G → C ( X, T ) = U ( M ( C ( X ))) defined by η ( g )( χ ) := χ ( g ) satisfies(H2) because η ( G ) separates the points of X . If, in addition, η is strictlycontinuous, i.e., each compact subset of X is equicontinuous, then (H1) isalso satisfied, so that ( C ( X ) , η ) is a host algebra of G .(c) If A is any C ∗ -algebra and G := U ( M ( A )), endowed with the stricttopology, then the fact that G spans M ( A ) implies that η = id G satisfies(H1) and (H2), so that ( A , id G ) is a host algebra for G . Example 2.16 (Non-uniqueness of host algebras) Let G := Z . Then itscharacter group is b G ∼ = T , which is a compact group with respect to the topol-ogy of pointwise convergence. Since G is locally compact, C ∗ ( G ) ∼ = C ( T )is a full host algebra for G . Let A := C ([0 , η : Z → U ( M ( C ([0 , ∼ = C ([0 , , T ) by η ( n )( x ) := e πinx . Then η (1) : [0 , → T is a continuous bijection, which implies in particular that16 ( Z ) separates the points, so that (H2) holds. Further, Z is discrete, so that(H1) is trivially satisfied, and thus ( A , η ) is a host algebra. This host alge-bra is full because the representations of Z are in one-to-one correspondencewith Borel spectral measures on T and η (1) is a Borel isomorphism. Note inparticular that the full host algebra A is not unital, although G is a discretegroup. Remark 2.17
Let G be a Lie group, A a C ∗ -algebra and η : G → U ( M ( A ))a group homomorphism. We then obtain left and right actions of G on A by isometries. Let A ∞ ⊆ A denote the set of all elements for which theorbit maps of these actions are smooth. The space A ∞ l of smooth vectors forthe left action η l is a right ideal, the set A ∞ r of smooth vectors for the rightaction η r is a left ideal and both are exchanged by the involution. Hencetheir intersection A ∞ is a ∗ -subalgebra on which G acts by multipliers. Definition 2.18
Let G be a Lie group and A a C ∗ -algebra. We say that ahomomorphism γ : G → U ( M ( A )) is strictly smooth if the ∗ -subalgebra A ∞ of smooth vectors for the left and right action of G on A is dense.If this is the case, then the maximal C ∗ -subalgebra A c of A on which G acts continuously is dense, hence coincides with A , so that η is in particularstrictly continuous (cf. Remark 2.14(c)). Proposition 2.19
Let ( S, η, W ) be a host semigroup of the Lie group G and α a G -invariant locally bounded absolute value on S . Then η induces astrictly smooth homomorphism e η : G → U ( M ( C ∗ ( S, α ))) determined uniquelyby e η ( g )( η α ( s )) = η α ( gs ) for g ∈ G , s ∈ S , and ( C ∗ ( S, α ) , e η ) is a host algebrafor G . Proof.
The existence of e η follows from Proposition 2.3, since U ( M ( S ))acts by unitary multipliers on C ∗ ( S, α ). That e η defines a strictly smoothmultiplier action of G on C ∗ ( S, α ) follows from the relation e η ( g ) η α ( s ) = η α ( gs ), which implies that η α ( S ) consists of smooth vectors for G because η α : S → C ∗ ( S, α ) is a holomorphic map. Hence e η is strictly smooth.To see that e η defines a host algebra, let ( π i , H ), i = 1 ,
2, be two rep-resentations of C ∗ ( S, α ) with e π ◦ e η = e π ◦ e η . For each x ∈ W we thenhave e π ◦ e η x = e π ◦ e η x : R → U ( H ) . Now e π i ◦ η x : C + → B ( H ), i = 1 ,
2, are two continuous representations ofthe involutive semigroup C + which are holomorphic on C and coincide17n R , so that [Ne99], Lemma XI.2.2, implies that they are equal. Hence π ◦ η x = π ◦ η x for each x ∈ W , so that π − π : S → B ( H ) is a holomorphicfunction vanishing on all sets γ Sx ( C ), and now (HS3) leads to π = π . C ∗ -algebras In the preceding section we have seen that we can associate to each hostsemigroup S and any G -invariant locally bounded absolute value α on S ahost algebra C ∗ ( S, α ) for G . In this section we slightly change our perspectiveand ask for properties of a homomorphism η : G → U ( M ( A )) which arecharacteristic for a host algebras of the form A = C ∗ ( S, α ). This will lead usto the momentum set I η ⊆ L( G ) ′ of the pair ( A , η ). We shall see in particularthat the weak- ∗ -closed convex Ad ∗ ( G )-invariant set I η tells us for which openinvariant cones W ⊆ L( G ) there might be a corresponding host semigroup. If G is a topological group and A a C ∗ -algebra, we also call a strictly con-tinuous homomorphism η : G → U ( M ( A )) a strictly continuous multiplieraction of G on A . Remark 3.1 (a) If G is a locally compact group, A is a C ∗ -algebra and thehomomorphism γ : G → U ( M ( A )) is strictly continuous, then integrationyields a morphism γ : L ( G ) → M ( A ) , γ ( f ) := Z G f ( g ) γ ( g ) dg of Banach- ∗ -algebras, so that the universal property of the C ∗ -algebra C ∗ ( G ),the enveloping C ∗ -algebra of L ( G ), implies the existence of a correspond-ing morphism of C ∗ -algebras e γ : C ∗ ( G ) → M ( A ) for which e γ ( C ∗ ( G )) A ⊇ γ ( L ( G )) A is dense in A (use an approximate identity in L ( G ) and thestrict continuity of the action of G ) and we have e γ ◦ η = γ. If, conversely, α : C ∗ ( G ) → M ( A ) is a morphism of C ∗ -algebras for which α ( C ∗ ( G )) A is dense in A , then Proposition 8.3 in the appendix implies that α extends to strictly continuous morphism e α : M ( C ∗ ( G )) → M ( A ). Hence18 := e α ◦ η : G → U ( M ( A )) is a strictly continuous homomorphism with e γ = e α .We conclude that strictly continuous multiplier actions of G on a C ∗ -algebra A are in one-to-one correspondence with morphisms α : C ∗ ( G ) → M ( A ) for which α ( C ∗ ( G )) A is dense in A . Example 3.2
Let A := K ( H ) be the C ∗ -algebra of compact operators onthe complex Hilbert space H . Then M ( A ) ∼ = B ( H ) is the algebra of allbounded operators on H . Hence U ( M ( A )) ∼ = U ( H ) is the unitary group of H and the strict topology on this group coincides with the strong operatortopology because if U ( H ) carries the strong operator topology, the closedsubalgebra K ( H ) c contains finite rank operators, hence coincides with K ( H ).Therefore a strictly continuous multiplier action of a topological group G on K ( H ) is the same as a continuous unitary representation on H . Remark 3.3 If G is a finite dimensional Lie group and η : G → U ( M ( A ))is strictly continuous, then it is also strictly smooth. In fact, there existsa sequence ( δ n ) n ∈ N ∈ C ∞ c ( G, R ) which is an approximate identity in L ( G ).Then each a ∈ A is the norm limit of the elements η ( δ n ) aη ( δ n ) ∈ A ∞ . Momentum sets of unitary representations
Definition 3.4
Let ( π, H ) be a continuous unitary representation of the Liegroup G on the Hilbert space H . We write H ∞ ⊆ H for the subspace ofsmooth vectors and note that this is a linear subspace on which we have aderived representation dπ of the Lie algebra g = L( G ). We call the represen-tation ( π, H ) smooth if H ∞ is dense in H .(a) Let P ( H ∞ ) = { [ v ] := C v : 0 = v ∈ H ∞ } denote the projective spaceof the subspace H ∞ of smooth vectors. The mapΦ π : P ( H ∞ ) → g ′ with Φ π ([ v ])( x ) = 1 i h dπ ( x ) .v, v ih v, v i is called the momentum map of the unitary representation π . The right handside is well defined because it only depends on [ v ] = C v . The operator i · dπ ( x ) is symmetric so that the right hand side is real, and since v is asmooth vector, it defines a continuous linear functional on g .(b) The weak- ∗ -closed convex hull I π ⊆ g ′ of the image of Φ π is called the convex momentum set of π . 19 emark 3.5 If π : A → B ( H ) is a non-degenerate representation of a C ∗ -algebra and η : G → U ( M ( A )) a strictly smooth multiplier action, then thecorresponding representation e π ◦ η of G on H has a dense space H ∞ of smoothvectors because the dense subspace spanned by π ( A ∞ ) H consists of smoothvectors for G . Lemma 3.6
Let G be a Lie group with exponential function and ( π, H ) aunitary representation with a dense space H ∞ of smooth vectors. Then, foreach x ∈ L ( G ) , the unbounded operator dπ ( x ) : H ∞ → H , dπ ( x ) v := ddt t =0 π (exp G ( tx )) .v is essentially skewadjoint and its closure dπ ( x ) is the infinitesimal generatorof the unitary one-parameter group π x := π ◦ γ x , γ x ( t ) = exp G ( tx ) . Proof.
Since the dense subspace H ∞ is invariant under π ( γ x ( R )), theassertion follows from [RS75], Thm. VIII.10. Lemma 3.7 If ( π, H ) is a smooth unitary representation of the Lie group G with exponential function, x ∈ L ( G ) , π x := π ◦ γ x and A x := − idπ x (1) thecorresponding selfadjoint operator, then inf Spec( A x ) = inf h I π , x i . Proof.
For m ( x ) := inf h I π , x i ∈ R ∪ {−∞} , we have h A x .v, v i ≥ m ( x ) h v, v i for each v ∈ H ∞ , and since the graph of the operator − i · dπ ( x ) on H ∞ in dense in the graphof A x (Lemma 3.6), h A x .v, v i ≥ m ( x ) h v, v i holds for each v in the domainof A x . This shows that inf Spec( A x ) ≥ m ( x ). The converse inequality holdstrivially. Problem 3.8
Let ( π, H ) be a unitary representation of the Lie group G on H . If v is a smooth vector, then the function π v ( g ) := h g.v, v i is smooth.In [Ne99], Prop. X.6.4 it is shown that the converse also holds if G is finitedimensional. Does this result generalize to infinite dimensional Lie groups?20 olomorphic extension of multiplier actions Definition 3.9
Let η : G → U ( M ( A )) be a strictly smooth multiplier actionof G on the C ∗ -algebra A and g = L( G ).We write S ( A ) for the set of states of A . Since each state of A is ofthe form ϕ ( a ) = π v ( a ) := h π ( a ) .v, v i for a unit vector v ∈ H and a non-degenerate representation ( π, H ) of A , there exists a canonical extension e ϕ := e π v to a state of M ( A ).We call a state ϕ of A η -smooth if e ϕ ◦ η is smooth and write S ( A ) ∞ forthe set of η -smooth states of A . We now have a momentum map Ψ η : S ( A ) ∞ → g ′ , Ψ η ( ϕ ) = 1 i d ( e ϕ ◦ η )( ) . Since S ( A ) ∞ is a convex set, the weak- ∗ -closure I η := Ψ η ( S ( A ) ∞ ) ⊆ g ′ also is a convex subset of g ′ ; called the momentum set of ( A , η ). Proposition 3.10
Let η : G → U ( M ( A )) be a strictly smooth multiplierrepresentation, I η ⊆ g ′ its momentum set and m ∈ R . Then the followingare equivalent for x ∈ g : (1) If η x ( t ) := η (exp G ( tx )) , then the corresponding homomorphism of C ∗ -algebras e η x : C ∗ ( R ) ∼ = C ( R ) → M ( A ) factors through the quotientalgebra C ([ m, ∞ [) . (2) I η ( x ) ≥ m . (3) η x extends to a strictly continuous homomorphism b η x : C + → M ( A ) of involutive semigroups which is holomorphic on C and satisfies k b η x ( z ) k ≤ e − m Im z .If these conditions are satisfied, then k b η x ( a + ib ) k = e − b · inf I η ( x ) . Proof. (1) ⇔ (2): Let ( π, H ) be a universal representation of A , i.e.,each state of A is of the form π v ( a ) = h π ( a ) .v, v i for some unit vector v ∈ H .Then e π ◦ η is a smooth representation of G (Remark 3.5) and η x also definesa continuous unitary representation π x := e π ◦ η x of R on H .21or any smooth unit vector v ∈ H ∞ and the corresponding smooth state π v we then obtain with ( e π v ◦ η )(exp G ( tx )) = π x ( t ):Ψ η ( π v )( x ) = − i · d ( e π v ◦ η )( x ) = − i h dπ ( x ) .v, v i = Φ π ([ v ])( x ) , which leads to I π ( x ) ⊆ I η ( x ). In view of [Ne99], Prop. X.6.4, the smooth-ness of a state π v implies the smoothness of v for π x , so that I η ( x ) ⊆ I π x .Lemma 3.7 now implies that inf I π x = inf h I π , x i = inf I π ( x ), so that we arriveat inf I π ( x ) = inf I η ( x ) = inf I π x . A simple argument with the spectral measure of the unitary one-parametergroup π x shows that the kernel of the corresponding representation b π x of C ∗ ( R ) ∼ = C ( R ) contains the ideal I m := { f ∈ C ( R ) : supp( f ) ⊆ ] − ∞ , m [ } if and only if m ≤ I π x , which means that it factors through the quotientalgebra C ( R ) /I m ∼ = C ([ m, ∞ [) . If this is the case, then the image of b π x liesin the multiplier algebra M ( A ) ∼ = { T ∈ B ( H ) : T A + A T ⊆ A} (Proposi-tion 8.3). This proves the equivalence of (1) and (2).(1) ⇒ (3): First we consider the map γ : C + → C b ([ m, ∞ [) = M ( C ([ m, ∞ [)) , γ ( z )( t ) := e izt . Then k γ ( z ) k = e − m Im z , so that γ is locally bounded. Since the strict topologyon bounded subsets of C b ([ m, ∞ [) coincides with the compact open topology(cf. [Br77], Lemma 3.5), the strict continuity of γ follows from the continuityof the map β : C → C b ([ m, ∞ [) , β ( z )( t ) := e izt with respect to the compactopen topology. That γ is holomorphic on the open upper halfplane C is aconsequence of Example 1.7(c). Now (3) follows by composing the strictlycontinuous extension M ( C ([ m, ∞ [)) ∼ = C b ([ m, ∞ [) → M ( A ) with γ .(3) ⇒ (2): By definition, b η x induces a morphism β : C ∗ ( C , α ) → M ( A ),where α ( z ) := e − m Im z . Since b η x is strictly continuous, β ( C ) A is dense in A , so that β extends to a strictly continuous morphism b β : M ( C ∗ ( C , α )) ∼ = C b ([ m, ∞ [) → M ( A ) with b β ( γ ( t )) = η x ( t ) for t ∈ R . Therefore the homo-morphism e η x : C ∗ ( R ) → M ( A ) factors through the quotient C b ([ m, ∞ [).22 emark 3.11 (a) As the example A = C ([ m, ∞ [) shows, the map b η x neednot be norm continuous because the natural map γ : C + → C b ([ m, ∞ [) , γ ( z )( t ) = e itz is not norm continuous at the boundary R = ∂ C + .(b) Assume that the conditions of Proposition 3.10 are for the element x ∈ g . Let B := M ( A ) c denote the C ∗ -subalgebra consisting of all elementson which G acts continuously by multipliers from the left and the right. Then η x ( R ) B + B η x ( R ) ⊆ B implies that η x induces a strictly continuous morphism η B x : R → U ( M ( B )).Since the induced homomorphism C ∗ ( R ) → M ( A ) factors through C ([ m, ∞ [), the same holds for the corresponding morphism C ∗ ( R ) → M ( B ).From that we conclude that we even obtain a strictly continuous morphism C + → M ( B ) which is holomorphic on C . Proposition 3.12
Let η : G → U ( M ( A )) be the strictly smooth multiplieraction defined by a host algebra of G obtained from a host semigroup ( S, η, W ) for which the map Exp : W → S, x b η x ( i ) is continuous. Then s : W → R , s ( x ) := − inf h I η , x i is a locally bounded function on W . Proof.
For each x ∈ W , the homomorphism η x : R → U ( M ( C ∗ ( S, α )))extends to a homomorphism of involutive semigroups b η x : C + → M ( C ∗ ( S, α ))which is holomorphic on C and comes from a smooth multiplier action of C + on S (Definition 2.7). We also have k b η x ( z ) k ≤ α ( γ x ( z )) for z ∈ C . From Example 1.7(c) we now derive that k b η x ( z ) k = e − m x Im z for some m x ∈ R and in particular that b η x is locally bounded on C + . Therefore the continuityof the corresponding multiplier action of C + on S implies the strict continuityof the corresponding multiplier action on C ∗ ( S, α ). Now Proposition 3.10(3)tells us that inf I η ( x ) = m x , so that s ( x ) = − m x . Further, e s ( x ) = e − m x = k b η x ( i ) k = k η α ( b η x ( i )) k ≤ α (Exp( x ))shows that s is locally bounded on the open cone W .23 Convex functions on infinite dimensionaldomains
In Proposition 3.12 we have seen how host algebras of a Lie group G comingfrom host semigroups lead to weak- ∗ -closed convex subsets I η of the dual L( G ) ′ of the locally convex Lie algebra L( G ) with the property that thesupport function x
7→ − inf h I η , x i is locally bounded on some open convexcone in L( G ).In this section we therefore take a closer look at weak- ∗ -closed convexsubsets C of the dual V ′ of a locally convex space V . We are in particularinterested in conditions for the cone B ( C ) = { v ∈ V : inf h C, v i > −∞} tohave non-empty interior and the corresponding support function s C ( v ) := − inf h C, v i to be locally bounded on B ( C ) . We start with a general discus-sion of convex sets and then turn to convex functions in the second subsection. Convex subsets of locally convex spaces
Proposition 4.1
Let ∅ 6 = C ⊆ V be a closed convex set in the topologicalvector space V . (1) lim( C ) := { v ∈ V : C + v ⊆ C } is a closed convex cone, called therecession cone of C . (2) lim( C ) = { v ∈ V : v = lim n →∞ t n c n , c n ∈ C, t n → , t n ≥ } . (3) If c ∈ C and x ∈ V satisfy c + R + x ⊆ C , then x ∈ lim( C ) . (4) If C is bounded, then lim( C ) = { } . Proof. (1) The closedness of lim( C ) is an immediate consequence of theclosedness of C .(2) If c ∈ C and x ∈ lim( C ), then c + nx ∈ C for n ∈ N and n ( c + nx ) → x. If, conversely, x = lim n →∞ t n c n with t n → t n ≥ c, c n ∈ C , then(1 − t n ) c + t n c n → c + x ∈ C = C implies that C + x ⊆ C , i.e. x ∈ lim( C ).(3) In view of (2), this follows from n ( c + nx ) → x .(4) If C is bounded, each continuous linear functional f : V → R isbounded on C . For each x ∈ lim( C ) the relation C + N x ⊆ C then leads to f ( x ) = 0. Since V ′ separates the points of V , we obtain x = 0.24 efinition 4.2 Let V be a locally convex space and C ⊆ V ′ a subset. Weput B ( C ) := { v ∈ V : inf h C, v i > −∞} and C ⋆ := { v ∈ V : h C, v i ⊆ R + } . Then C ⋆ ⊆ B ( C ) are convex cones and C ⋆ is called the dual cone of C . If C is a cone, then B ( C ) = C ⋆ .Let C ⊆ V ′ be a weak- ∗ -closed convex subset. As a consequence ofthe Hahn–Banach Separation Theorem, there exists for each element α ∈ V ′ \ C some x ∈ V (the dual of V ′ endowed with the weak- ∗ -topology) with α ( v ) < inf h C, v i . Then v ∈ B ( C ), and we thus obtain C = { α ∈ V ′ : ( ∀ v ∈ B ( C )) α ( v ) ≥ inf h C, v i} , (2)which permits us to reconstruct C from its support function s C ( v ) = − inf h C, v i on V . Lemma 4.3
For a non-empty weak- ∗ -closed convex subset C ⊆ V ′ , the fol-lowing assertions hold: (i) B ( C ) is a convex cone satisfying B ( C ) ⋆ = lim( C ) . (ii) If B ( C ) has non-empty interior, then B ( C ) has the same interior as lim( C ) ⋆ . (iii) If C is a cone, then B ( C ) = C ⋆ has non-empty interior if and only if C has a weak- ∗ -compact equicontinuous base. Proof. (i) The relation C + lim( C ) = C implies that every element in B ( C ) is non-negative on lim( C ), i.e. lim( C ) ⊆ B ( C ) ⋆ .Using (2), we see that for x ∈ B ( C ) ⋆ , c ∈ C and f ∈ B ( C ), we have f ( x + c ) ≥ f ( c ) ≥ inf f ( C ) , so that x + C ⊆ C . This proves B ( C ) ⋆ ⊆ lim( C )and hence equality.(ii) From the Hahn–Banach–Separation Theorem, we further derive B ( C ) = ( B ( C ) ⋆ ) ⋆ = lim( C ) ⋆ . If B ( C ) has non-empty interior, it coincideswith the interior of its closure ([Bou07], Cor. II.2.6.1).(iii) If C has an equicontinuous weak- ∗ -compact base K and x ∈ V satisfies h K, x i > ε , then ε b K (where b K denotes the polar of K ) is a 0-neighborhood in V with x + ε b K ⊆ C ⋆ , showing that C ⋆ has interior points.25f, conversely, C ⋆ has an interior point x and U is a convex symmetric0-neighborhood with x + U ⊆ C ⋆ , then the polar set b U is a weak- ∗ -compactequicontinuous subset containing K := { α ∈ C : α ( x ) = 1 } . Therefore K isweak- ∗ -compact and equicontinuous with C = R + K and 0 K , i.e., K is abase of C .If B ( C ) has interior points, the following proposition shows that we canreconstruct C from the values of s C on the open set B ( C ) . Proposition 4.4
Let V be a locally convex space and C ⊆ V ′ be a weak- ∗ -closed convex subset for which the cone B ( C ) has interior points. Then C = { α ∈ V ′ : ( ∀ x ∈ B ( C ) ) α ( x ) ≥ inf h C, x i} . Proof.
Let D := { α ∈ V ′ : ( ∀ x ∈ B ( C ) ) α ( x ) ≥ inf h C, x i} . Then wehave C ⊆ D and both are weak- ∗ -closed convex sets. If α ∈ D \ C , then (2)implies the existence of some x ∈ B ( C ) with α ( x ) < inf h C, x i .Let x ∈ B ( C ) . Then, for each t ∈ ]0 , x t := (1 − t ) x + tx iscontained in B ( C ) , so that α ( x t ) ≥ inf h C, x t i . Now F : [0 , → R , F ( t ) := − inf h C, x t i is a lower semicontinuous convex function on a real interval,hence continuous ([Ne99], Cor. V.3.3). Thereforeinf h C, x i = lim t → inf h C, x t i ≤ lim t → α ( x t ) = α ( x ) , and we get α ( x ) ≥ inf h C, x i . This contradiction implies that C = D .The following lemma is obvious: Lemma 4.5
For a weak- ∗ -closed convex subset C ⊆ V ′ the following areequivalent (1) C is weak- ∗ -bounded. (2) B ( C ) = V . (3) The polar set b C := { v ∈ V : |h C, v i| ≤ } is absorbing. onvex functions on domains in locally convex spaces Definition 4.6
Let X be a topological space. We say that a real-valuedfunction f : X → R ∞ := R ∪ {∞} is lower semicontinuous if for each x ∈ X and c < f ( x ) there exists a neighborhood U of x with inf f ( U ) > c . Wecall f upper semicontinuous if for each x ∈ X and d > f ( x ) there exists aneighborhood U of x with sup f ( U ) < d . Remark 4.7
A function f : X → R ∞ is lower semicontinuous if and only ifits epigraph epi( f ) := { ( x, t ) ∈ X × R : f ( x ) ≤ t } is a closed subset of X × R . Proposition 4.8 (cf. [Bou07], Prop. II.2.21)
Let Ω ⊆ V be an open convexsubset and f : Ω → R a lower semicontinuous convex function. Then thefollowing are equivalent (1) f is continuous. (2) f is locally bounded. (3) f is bounded in a neighborhood of one point. (4) f is upper semicontinuous. Proof.
Since f is assumed to be lower semicontinuous, (1) and (4) areequivalent. Clearly, (1) ⇒ (2) ⇒ (3).(3) ⇒ (2): Suppose that f ≤ M holds on the open convex neighborhood U of c ∈ Ω. Let c ∈ Ω. Then there exists an element c ∈ Ω and 0 < t < c = (1 − t ) c + tc . Then (1 − t ) c + tU is an open subset of Ω containing c , and on this subset we havesup f (cid:0) (1 − t ) c + tU (cid:1) ≤ (1 − t ) f ( c ) + t sup f ( U ) < ∞ . Therefore f is locally bounded.(2) ⇒ (4): Let c ∈ Ω and U a closed convex 0-neighborhood in V with c + U ⊆ Ω on which f is bounded and d > f ( c ). Then for each t ∈ [0 , f ( c + tU ) = sup f ((1 − t ) c + t ( c + U )) ≤ (1 − t ) f ( c ) + t sup f ( c + U ) , so that for some t > f ( c + tU ) ≤ d . Therefore f isupper semicontinuous in c . 27 emark 4.9 Under the circumstances of Lemma 4.5, s C ( x ) := − inf h C, x i defines a convex function on all of V . In view of the preceding proposition,this function is locally bounded if and only if it is continuous, and this isequivalent to the polar set b C being a 0-neighborhood. The polar set b C is a barrel , i.e., a closed absolutely convex absorbing set. According to the BipolarTheorem, each barrel B coincides with the polar b C of its polar C := b B .A locally convex space V is said to be barrelled if all barrels in V are0-neighborhoods. In view of the preceding remarks, this means that all func-tions s C are continuous. The functions s C , b C a barrel, are precisely the lowersemicontinuous seminorms on V , so that V is barrelled if and only if all lowersemicontinuous seminorms are continuous (cf. [Bou07], § III.4.1).The following theorem extends this remark to general convex functions.
Theorem 4.10
Let V be a barrelled space, Ω ⊆ V an open convex set and f : Ω → R a lower semicontinuous function. Then f is continuous. Proof.
Pick x ∈ Ω and let U be a closed absolutely convex 0-neighborhoodwith x + U ⊆ Ω. We consider the set B := { v ∈ U : f ( x ± v ) ≤ f ( x ) + 1 } and claim that B is a barrel. Since f is lower semicontinuous, B is a closedconvex subset of U . Moreover, v ∈ B and λ ∈ R with | λ | ≤ λv ∈ B because f ( x ± λv ) ≤ conv { f ( x ± v ) } ≤ f ( x ) + 1 . We conclude that B is absolutely convex. To see that B is absorbing, let v ∈ U and observe that the lower semicontinuous function h ( t ) := f ( x + tv )on [ − ,
1] is continuous ([Ne99], Cor. V.3.3). Hence there exists a µ > µv ∈ B . This proves that B is a closed absolutely convex absorbingset, hence a barrel. Since V is barrelled, B is a 0-neighborhood, and thus f is bounded on a neighborhood of x . The continuity of f now follows fromProposition 4.8. Remark 4.11
For a locally convex space we have the following implicationsBanach ⇒ Fr´echet ⇒ Baire ⇒ barrelled , so that in particular all Fre´chet spaces are barrelled.28urthermore, each locally convex direct limit of barrelled spaces is bar-relled, which implies that there are barrelled spaces which are not Baire, f.i. V := R ( N ) , endowed with the finest locally convex topology, is such a space. Example 4.12 (A barrel which is not a 0-neighborhood) Let X = c , whichis a non-reflexive Banach space and V := ℓ its topological dual, endowedwith the weak- ∗ -topology. Then the closed unit ball B ⊆ V is a barrel whichis not a zero neighborhood because each 0-neighborhood contains a subspaceof finite codimension. Proposition 4.13
Let C ⊆ V ′ be a non-empty weak- ∗ -closed convex sub-set and x ∈ B ( C ) such that the support function s C is bounded on someneighborhood of x . Then the following assertions hold: (1) For each m ∈ R the subset C m := { α ∈ C : α ( x ) ≤ m } is equicontinuousand weak- ∗ -compact. (2) The function η ( x ) : C → R , η ( x )( α ) := α ( x ) is proper. (3) There exists an extreme point α ∈ C with α ( x ) = min h C, x i . (4) C is weak- ∗ -locally compact. Proof. (1) Pick a 0-neighborhood U ⊆ V for which s C is bounded on x + U by some constant M . Then h C, x + U i ≥ − M , and hence h C m , U i ≥ h C m , x + U i − m ≥ − M − m. This implies that s C m is bounded from below on U , and hence s C m is boundedfrom above on − U . Therefore the polar b C m contains a multiple of U ∩ − U ,hence is a neighborhood of 0. This is equivalent to C m being equicontinu-ous. Now the Banach–Alaoglu–Bourbaki Theorem implies that C m is weak- ∗ -compact because it is a closed subset of the polar set of a 0-neighborhoodin U .(2) follows immediately from (1).(3) Pick M > inf h C, x i . Then the weak- ∗ -compactness of C M implies theexistence of a minimal value m = min η ( x )( C ). Then C m := η ( x ) − ( m ) ∩ C is a weak- ∗ -compact convex set, so that the Krein–Milman Theorem impliesthe existence of an extreme point e of C m . Since C m is a face of C , e also isan extreme point of C .(4) For any α ∈ C , (1) implies that the set C α ( x )+1 is a compact neigh-borhood of α in C . Hence C is weak- ∗ -locally compact.29 emark 4.14 If C ⊆ V ′ is a weak- ∗ -closed convex subset which is locallycompact with respect to the weak- ∗ -topology, then its recession cone lim( C )is also locally compact because, for each α ∈ C , the subset α + lim( C ) of C isclosed. Therefore Exercise II.7.21(a) in [Bou07] implies that the cone lim( C )has a weak- ∗ -compact base K , but if K is not equicontinuous, this does notimply that the dual cone lim( C ) ⋆ = B ( C ) has interior points (Lemma 4.3). Remark 4.15 If C ⊆ V ′ is weak- ∗ -locally compact, then we consider thecommutative C ∗ -algebra A := C ( C ). Clearly, the map η : V → M ( C ( C )) ∼ = C b ( C ) , η ( x )( α ) := e iα ( x ) is a multiplier action of the abelian topological group V on C ( C ). Thisaction is strictly continuous if and only if η is continuous with respect to thecompact open topology on C b ( C ) (cf. [Br77], Lemma 3.5). If this is the case,then the additive map e η : V → C b ( C ) , e η ( x )( α ) = α ( x ) is also continuous withrespect to the compact open topology, and this is equivalent to the equicon-tinuity of each compact subset of C . Then η : V → C b ( C ) defines a strictlycontinuous multiplier action and Example 2.15(b) implies that ( C ( C ) , η ) isa host algebra for V . Problem 4.16 (a) Suppose that for some x ∈ V all sets C m := { α ∈ C : α ( x ) ≤ m } are equicontinuous, hence in particular weak- ∗ -compact. Then η ( x )( α ) := α ( x ) defines a proper function on C , showing that C is locally compact.The equicontinuity of the sets C m implies that the cone lim( C ) has anequicontinuous weak- ∗ -compact base, so that lim( C ) ⋆ has interior points(Lemma 4.3(iii)). Does this imply that B ( C ) has interior points and thatthe support function s C is bounded on some neighborhood of x ?(b) If, in addition, some function e − e η ( x ) , e η ( x )( α ) = α ( x ), is contained in C ( C ), then e η ( x ) is proper and bounded from below. Does the requirementthat η − ( C ( C )) ⊆ B ( C )has interior points imply that s C is bounded on some open set? Accordingto Theorem 4.10, this is the case if V is barrelled.30 Host C ∗ -algebras coming from tubes In this section we briefly take a closer look at the host algebras of a lo-cally convex space V , defined by a complex involutive semigroup of the type S = V + iW for an open convex cone W ⊆ V (Proposition 2.9). In viewof Remark 1.6(b) and the discussion in Example 1.7, any locally boundedabsolute value α on such a semigroup leads to the same C ∗ -algebra as anabsolute value of the form α C ( x + iy ) := e − inf h C,y i , where C ⊆ V ′ is a weak- ∗ -closed convex subset with W ⊆ B ( C ) whosesupport function is locally bounded on W . The following theorem providesa converse: Theorem 5.1
Let C ⊆ V ′ be a weak- ∗ -closed convex subset for which thecone B ( C ) has interior point and the support function s C is locally boundedon B ( C ) . Then the following assertions hold: (1) S := V + iB ( C ) is an open complex involutive subsemigroup of V C and α C ( x + iy ) := e − inf h C,y i is a locally bounded absolute value on S . (2) C is weak- ∗ -locally compact. (3) The map γ : S → C ( C ) , γ ( s )( f ) := e if ( s ) induces an isomorphism of C ∗ -algebras C ∗ ( S, α C ) → C ( C ) . (4) The homomorphism η : V → C ( C, T ) = U ( M ( C ( C ))) , η ( x )( f ) := e if ( x ) defines a host algebra of V with a strictly smooth multiplier action. Thecorresponding momentum set is I η = C . Proof. (1) is an immediate consequence of the definition.(2) This follows from Proposition 4.13.(3) For each element s = x + iy ∈ S , the continuous function γ ( s ) on C satisfies k γ ( s ) k = sup f ∈ C | e if ( s ) | = sup f ∈ C e − f ( y ) = e − inf h C,y i = α C ( s ) . ε >
0, the subset { f ∈ C : e − f ( y ) = | γ ( s )( f ) | ≥ ε } = { f ∈ C : f ( y ) ≤ − log ε } is weak- ∗ -compact (Proposition 4.13), so that γ ( s ) ∈ C ( C ).We thus obtain a morphism γ : S → C ( C ) of involutive semigroups with k γ ( s ) k ≤ α C ( s ). Hence γ is locally bounded, and to see that it is holomor-phic, it suffices to verify its holomorphy on the intersection of S with eachcomplex subspace E C ⊆ V C , where E ⊆ V is finite dimensional.Using that the C ∗ -algebra C ( C ) has a realization as a closed subalgebraof some algebra of the form B ( H ), we first use [Ne99], Thm. VI.2.3, to seethat ρ := γ | iB ( C ) ∩ E C is a norm continuous morphisms of semigroups. Further,Proposition VI.3.2 in [Ne99] implies that ρ extends to a unique holomorphichomomorphism b ρ : S ∩ E C → B ( H ), but since iB ( C ) ∩ E is totally real in S ∩ E C , the values of the unique holomorphic extension b ρ also lie in the closedsubspace C ( C ) of B ( H ). For each f ∈ C , the function b ρ ( s )( f ) on S ∩ E C is the unique holomorphic extension of the function η ( s )( f ) = ρ ( s )( f ), andfrom the holomorphy of S → C , s γ ( s )( f ) we derive that b η = γ | S ∩ E C . Thisproves that γ is holomorphic on S ∩ E C , and hence that γ is holomorphic.Now the universal property of the C ∗ -algebra C ∗ ( S, α C ) leads to a uniquemorphism b γ : C ∗ ( S, α C ) → C ( C ) of C ∗ -algebras with b γ ◦ η = γ (Theo-rem 1.5).To see that b γ is injective, we recall that the characters of C ∗ ( S, α C ) sep-arate the points, so that it suffices to show that they are all of the form f b γ ( f )( f ) for some f ∈ C . Any non-zero character χ of C ∗ ( S, α C ) isuniquely determined by the holomorphic character χ S := χ ◦ η : S → C . Weclaim that χ S ( S ) ⊆ C × . Indeed, χ S ( s ) = 0 implies χ S ( s + S ) = { } , so that χ S = 0 would follow by analytic continuation. Now χ S ( S ) ⊆ C × shows thatwe have a corresponding smooth character χ V : V → T , obtained from thesmooth multiplier action of V on S . We further derive that there exists some β ∈ V ′ with χ S ( s ) = e iβ ( s ) for s ∈ S = V + iB ( C ) . Since any morphism of C ∗ -algebras is contractive, we get | χ S ( x + iy ) | = e − β ( y ) ≤ α C ( s ) = e − inf h C,y i , i.e., β ( y ) ≥ inf h C, y i for x ∈ B ( C ) . Now we apply Proposition 4.4 to obtain β ∈ C . If, conversely, β ∈ C , then e iβ defines an α C -bounded holomorphic32haracter of S , and the universal property of C ∗ ( S, α C ) implies that thischaracter extends to a character of C ∗ ( S, α C ). These arguments show thatthe characters of C ∗ ( S, α C ), resp., the α C -bounded holomorphic characters of S , are of the form s γ ( s )( β ) for some β ∈ C . As we have already observedabove, this implies that b γ is injective, hence an isometric embedding ([Dix64],Cor. I.8.3).In view of the Stone–Weierstraß Theorem, the fact that the functions in γ ( S ) have no zeros and separate the points of C implies that b γ has denseimage. We know already that b γ is isometric, so that its range is closed.Therefore b γ is an isomorphism of C ∗ -algebras.(4) First we combine Proposition 2.19 with Example 2.10 to see that( C ( C ) , η ) is a host algebra of V with strictly smooth multiplier action.To calculate the corresponding momentum set, we first recall from (3)that S α = C , so that the character χ f ( ξ ) := ξ ( f ) defined by f ∈ C definesa smooth state of C ( C ) with e χ f ( η ( x )) = e if ( x ) , which leads to Ψ η ( χ f ) = f ,and thus C ⊆ I η . On the other hand, Proposition 3.10 shows that for each y ∈ B ( C ) , we have e − inf I η ( y ) = k b η y ( i ) k = α ( iy ) = e − inf h C,y i , so that inf h C, y i = inf I η ( y ), which leads to I η ⊆ C (Proposition 4.4).The preceding theorem implies in particular that each weak- ∗ -closed con-vex subset C ⊆ V ′ whose support function is bounded on some open subsetactually occurs as the momentum set of some host algebra of V . Conversely,we have seen in Example 1.7(c) that all host algebras defined by complexsemigroups of the form V + iW , W an open convex cone in V , are of thisform. We thus obtain a complete picture for the case where G = ( V, +) isthe additive group of a locally convex space. Remark 5.2
One can also develop a holomorphic representation theory oftubes of the form V + iW by starting with representations of the cone iW ∼ = W by selfadjoint operators. This program has been carried out in greatgenerality by H. Gl¨ockner in [Gl03] (cf. also [Gl00]). In the preceding section we have described all host algebras of locally con-vex spaces defined by complex host semigroups. In the non-commutative33ase this turns out to be much harder. However, using [Ne99], we also ob-tain a complete picture for finite dimensional groups. We shall take up theinvestigation of the non-abelian infinite dimensional case in the future.We write S ∞ ( G ) for the set of smooth states of G , i.e., the set of smoothpositive definite functions normalized by ϕ ( ) = 1. We consider S ∞ ( G ) as aconvex subset of the set S ( G ) of continuous states of G , which in turn canbe identified with the state space S ( C ∗ ( G )) of the group algebra C ∗ ( G ). Werecall the following result from [Ne99], Prop. X.6.17. Proposition 6.1
Let C ⊆ L ( G ) ∗ be a closed convex invariant subset and ev : S ∞ ( G ) → L ( G ) ∗ , ϕ i dϕ ( ) . Then the annihilator I C := ev − ( C ) ⊥ is an ideal of C ∗ ( G ) . The non-degeneraterepresentations of the quotient algebra C ∗ ( G ) C := C ∗ ( G ) /I C correspond tothose continuous unitary representations ( π, H ) of G satisfying I π ⊆ C . Theorem 6.2
Let G be a connected finite dimensional Lie group, ( S, η S , W ) a host semigroup of G and α a locally bounded absolute value on S . Thenthe following assertions hold: (a) The host algebra ( C ∗ ( S, α ) , η ) is a quotient of C ∗ ( G ) . (b) If, in addition, G acts on C ∗ ( S, α ) with discrete kernel and the polar map G × W → S, ( g, x ) g Exp x is a diffeomorphism, then C ∗ ( S, α ) ∼ = C ∗ ( G ) I η . Proof. (a) Let ( π, H ) be the universal representation of C ∗ ( S, α ). Thenwe have a holomorphic representation b π : S → B ( H ) whose image generatesthe C ∗ -algebra B := π ( C ∗ ( S, α )) ∼ = C ∗ ( S, α ) (Theorem 1.5).Let π G : G → U ( H ) denote the corresponding unitary representation of G and b π G the associated representation of C ∗ ( G ). Since G acts smoothly bymultipliers on S , we obtain a continuous multiplier action of G on B , andthis leads to b π G ( C ∗ ( G )) B ⊆ B , where the left hand side is dense in B .On the other hand, G also acts continuously by unitary multipliers on C ∗ ( G ), hence on b π G ( C ∗ ( G )). For each x ∈ W , we now obtain a mor-phism of C ∗ -algebras e π x : C ∗ ( R ) → M ( b π G ( C ∗ ( G ))) which factors throughsome quotient C ([ m, ∞ [). This implies that b π (Exp( x )) ∈ M ( b π G ( C ∗ ( G ))),and from that we obtain b π ( S ) ⊆ M ( b π G ( C ∗ ( G ))) by analytic continuation34nd (HS3) in the definition of a host semigroup. This in turn leads to B b π G ( C ∗ ( G )) ⊆ b π G ( C ∗ ( G )). Since e π x ( C ([ m, ∞ [)) b π G ( C ∗ ( G )) is dense in b π G ( C ∗ ( G )), we see that B b π G ( C ∗ ( G )) spans a dense subspace of b π G ( C ∗ ( G )).We now arrive that B = span( b π G ( C ∗ ( G )) B ) = b π G ( C ∗ ( G )) , and this proves (a).(b) From the proof of (1) ⇔ (2) in Proposition 3.10 we recall that I π = I η .Let C := I η . Then the ideal I C of C ∗ ( G ) annihilates all states of the form π v , v ∈ H , so that I C ⊆ ker b π G .By assumption, dπ is a faithful representation of L( G ), so that { } =ker dπ = I ⊥ π implies that I π spans the dual space L( G ) ′ . Therefore theopen cone W ⊆ B ( I π ) satisfies Spec(ad x ) ⊆ i R for each x ∈ W ([Ne99],Prop. VII.3.4(b)), i.e., W is a weakly elliptic cone ([Ne99], Def. XI.1.11).The construction in Section XI.1 in [Ne99] now leads to a complex involutivesemigroup Γ G ( W ) for which the polar map G × W → Γ G ( W ) , ( g, x ) g Exp( x )is a diffeomorphism. According to the Holomorphic Extension Theorem([Ne99], XI.2.3), each unitary representation ( ρ, K ) of G with I ρ ⊆ C = I η extends via ρ : Γ G ( W ) → b π ( S ) ⊆ B ⊆ B ( H ) , g Exp x π ( g ) e i · dπ ( x ) to a holomorphic representation b ρ of Γ G ( W ) with k b ρ ( g Exp x ) k = e − inf h I ρ ,x i ≤ e − inf h C,x i = e − inf h I η ,x i ≤ α ( g Exp x ) , hence to a representation of B . In view of (a), these representations separatethe points of C ∗ ( G ) C , which implies that I C = ker b π G . We finally obtain C ∗ ( S, α ) ∼ = B ∼ = C ∗ ( G ) C . Remark 6.3 If G = V is a finite dimensional vector space, W ⊆ V an openconvex cone, and C ⊆ V ′ a closed convex subset, then C ∗ ( G ) ∼ = C ( b G ) ∼ = C ( V ′ ), and the definition of C ∗ ( G ) C , implies that C ∗ ( G ) C ∼ = C ( C ). Wehave already seen in Section 5 that this is C ∗ ( S, α ) for S = V + iW and α ( x + iy ) = e − inf h C,y i . 35 Open Problems
Problem 7.1 (Invariant convex geometry of Lie algebras) Study open in-variant convex cones in the Lie algebra L( G ) of an infinite dimensional Liegroup G . Here are some concrete problems: • Does it have any consequence for the spectrum of ad x if x is containedin an open invariant convex cone W not containing affine lines? Whatcan be said about the stabilizer of x in G and its action on the Liealgebra L( G )? In the finite-dimensional case it acts like a compactgroup and, consequently, ad x is semisimple with Spec(ad x ) ⊆ i R (cf.[Ne99]). • Develop a structure theory for coadjoint orbits O f := Ad ∗ ( G ) .f ⊆ L( G ) ′ for which the weak- ∗ -closed convex hull C f has the propertythat B ( C f ) has interior points and the support function s C f is locallybounded on the interior. For any such orbit which separates the pointsof L( G ) (which can always be arranged after factorization of a closedideal), the open cone B ( C f ) does not contain affine lines. It is a nat-ural question under which circumstances the coadjoint orbit is closed.The geometric setup leads to the alternative that either O f consists ofextreme points of its weak- ∗ -closed convex hull or not, where the lattercase does not arise for closed orbits in the finite-dimensional case (cf.Section VIII.1 in [Ne99]). Remark 7.2 If x ∈ B ( C ) and f ∈ C is a unique minimum of the function e η ( x )( α ) = α ( x ) on C , then the stabilizer of x in G is contained in thestabilizer of G f and it also preserves all weak- ∗ -compact subsets C m = { α ∈ C : α ( x ) ≤ m } of L( G ) ′ . This situation should lead to interesting geometric structures onthe coadjoint orbit O f , such as weak K¨ahler structures. Example 7.3
Let A be a unital C ∗ -algebra and G = U ( A ) its unitarygroup, considered as a Banach–Lie group. Then for each state ϕ ∈ S ( A ) thefunctional − iϕ : L( G ) = { a ∈ A : a ∗ = − a } = u ( A ) → R is real-valued. If ϕ is a pure state, i.e., an extreme point of S ( A ), then thecoadjoint orbit O − iϕ consists of extreme points of its weak- ∗ -closed convex36ull, which is the weak- ∗ -closed convex face of − iS ( A ), generated by − iϕ ([Ne02], Thm. III.1). Problem 7.4 (Holomorphic extensions of unitary representations)(1) Suppose that ( π, H ) is a unitary representation of the infinite dimen-sional Lie group G for which the subspace H ∞ of smooth vectors isdense and B ( I π ) has interior points. Let x ∈ B ( I π ) . Is the selfadjointoperator e i · dπ ( x ) ∈ B ( H ) a smooth vectors for the multiplier action of G on B ( H )?(2) Is π ( G ) e i · dπ ( B ( I π ) ) a subsemigroup of B ( H )?(3) Suppose that there exists a host semigroup ( S, η, W ) for G for whichthe polar map G × W → S, ( g, x ) g Exp( x ) is a diffeomorphism.Assume that W ⊆ B ( I π ) for some smooth unitary representation ( π, H )of G . Is the map b π : S → B ( H ) , g Exp( x ) π ( g ) e i · dπ ( x ) a holomorphicrepresentation?(4) If L( G ) contains a dense locally finite subalgebra, many of the argu-ments seem to be reducible to the finite dimensional situation. Problem 7.5 (Existence of complex semigroups) Let G be a Banach–Liegroup (or locally exponential) and W ⊆ L( G ) an open convex cone satis-fying Spec(ad x ) ⊆ i R for each x ∈ W . We assume that G has a faithfuluniversal complexification η : G → G C (which is locally exponential if G isnot Banach). Is it true that G exp( iW ) a subsemigroup of G C ? For someinteresting examples of such semigroups we refer to [Ne01]. The following results are used in our discussion of general host algebras oftopological groups.
Theorem 8.1 ([Pa94], Th. 5.2.2)
Let A be a Banach algebra with boundedleft approximate identity and T : A → B ( X ) a continuous representationof A on the Banach space X . Then for each y ∈ span( T ( A ) X ) there areelements a ∈ A and x ∈ X with y = T ( a ) x . orollary 8.2 If A and B are C ∗ -algebras and π : A → M ( B ) is a homo-morphism for which π ( A ) B is dense in B , then each element y ∈ B can bewritten as π ( a ) b for some a ∈ A and b ∈ B . Proposition 8.3
Let A and B be C ∗ -algebras. For each morphism α : A → M ( B ) of C ∗ -algebras for which α ( A ) B is dense in B , there exists a uniquemorphism of C ∗ -algebras e α : M ( A ) → M ( B ) extending α , and e α is strictlycontinuous. Proof.
The uniqueness of e α follows from the density of α ( A ) B in B and e α ( m ) α ( a ) b = α ( ma ) b for m ∈ M ( A ), a ∈ A , b ∈ B .For the existence, we realize B as a closed ∗ -subalgebra of some B ( H ) forwhich the representation on H is non-degenerate. Then M ( B ) ∼ = { T ∈ B ( H ) : T B + B T ⊆ B} (Examples 2.12) and we interprete α as a representation of A on H .We claim that α is non-degenerate. Indeed, if α ( A ) v = { } , then { } = h α ( A ) v, BHi = hB α ( A ) v, Hi implies B α ( A ) v = { } , and since B α ( A ) = ( α ( A ) B ) ∗ is dense in B , we obtain v = 0.As α is non-degenerate, there exists a unique extension e α : M ( A ) → B ( H )with e α ( m ) α ( a ) = α ( ma ) for m ∈ M ( A ) and a ∈ A . Then e α ( m ) α ( A ) B ⊆ α ( A ) B ⊆ B and B α ( A ) e α ( m ) ⊆ B α ( A ) ⊆ B , and the density of α ( A ) B , resp., B α ( A ) in B implies that e α ( M ( A )) ⊆ M ( B ).Suppose that c i → c strictly in M ( A ). Then we have for a ∈ A and b ∈ B the relation e α ( c i ) α ( a ) b = α ( c i a ) b → α ( ca ) b, and since B = α ( A ) B (Corollary 8.2), we get e α ( c i ) b → e α ( c ) b for each b ∈ B , showing that e α ( c i ) → e α ( c ) in the strict topology on M ( B ). Proposition 8.4
Let A be a C ∗ -algebra and M ( A ) its multiplier algebra.Then the following properties of a representation ( ρ, H ) of M ( A ) are equiv-alent (1) ρ | A is non-degenerate. ρ = e π for some non-degenerate representation of A . (3) The representation ρ is continuous with respect to the strict topologyon B and the strong operator topology on B ( H ) . Proof. (1) ⇒ (2): Let π := ρ | A and assume that this representation isnon-degenerate, so that it has an extension to a representation e π of M ( A )which is uniquely determined by e π ( m ) π ( a ) = π ( ma ) for m ∈ M ( A ) and a ∈ A . Since ρ also satisfies ρ ( m ) π ( a ) = ρ ( m ) ρ ( a ) = ρ ( ma ) = π ( ma ), weobtain ρ = e π .(2) ⇒ (3): Let ( b i ) i ∈ I be a net in B converging to some b ∈ B with respectto the strict topology, i.e., b i a → ba and ab i → ab for each a ∈ A . For each v ∈ H and a ∈ A we then have b i . ( a.v ) = ( b i a ) .v → ( ba ) .v = b. ( a.v ) and b ∗ i . ( a.v ) = ( a ∗ b i ) ∗ .v → ( a ∗ b ) ∗ .v = b ∗ . ( a.v ) . We conclude that b i .v → b.v and b ∗ i .v → b ∗ .v hold for all vectors v ∈ span( AH ), but Theorem 8.1 implies that H = AH , proving (3).(3) ⇒ (1): Let ( u i ) i ∈ I be an approximate identity in A . Then u i → in the strict topology on M ( A ). Hence we get for each v ∈ H the relation u i .v → v , showing that ρ | A is non-degenerate. References [Bou07] Bourbaki, N., “Espaces vectoriels topologiques. Chap.1 `a 5”,Springer-Verlag, Berlin, 2007[Br77] Brown, L. G.,
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