Induction and absorption of representations and amenability of Banach *-algebraic
aa r X i v : . [ m a t h . OA ] S e p INDUCTION AND ABSORPTION OF REPRESENTATIONSAND AMENABILITY OF BANACH *-ALGEBRAIC
DAMIÁN FERRARO
Abstract.
Given a Fell bundle B (saturated or not) over G and a closed subgroup H ⊂ G, we prove that any *-representation of the reduction B H can be induced to B . We observe that Exel-Ng’s reduced cross sectional C*-algebra C ∗ r ( B ) is universalfor the *-representations induced from B e = B { e } and construct a cross sectionalC*-algebra of B , C ∗ H ( B ) , that is universal for the *-representations induced from B H . We prove an absorption principle for C ∗ H ( B ) with respect to tensor products of *-representations of B and *-representations of G induced from H. Using this principlewe show, among other results, that given closed normal subgroups of
G, H ⊂ K, thereexists a quotient map q B KH : C ∗ K ( B ) → C ∗ H ( B ) which is a C*-isomorphism if and onlyif q B K KH : C ∗ ( B K ) → C ∗ H ( B K ) is a C*-isomorphism. We also prove the two conditionsabove hold if q GKH : C ∗ K ( G ) → C ∗ H ( G ) is a C*-isomorphism. All the constructions areperformed using Banach *-algebraic bundles having a strong approximate unit. Contents
Introduction 11. An addenda to Fell’s book on Banach *-algebraic bundles 42. Induction of *-representations 82.1. Integration and disintegration of *-representations 82.2. Positive *-representations and induction 102.3. Weak containment and Fell’s absorption principle 192.4. Induction in stages 273. Amenability 293.1. Amenability and reductions to normal subgroups 344. C*-completions of Banach *-algebraic bundles 384.1. Cross sectional bundles, C*-completions and induction 41References 48
Introduction
The (universal) crossed product of a C*-dynamical system (
A, G, α ) is a C*-algebra A ⋊ α G which is universal for the covariant pairs representing the system [10, 2.6].Given a closed subgroup H of the locally compact and Hausdorff (LCH) group G, one may construct a crossed product A ⋊ Hα G inducing a faithful *-representationfrom the crossed product A ⋊ α | H H associated to the restriction ( A, H, α | H ) (see, forexample, [10, 7.2]). The reduced crossed product of ( A, G, α ) , A ⋊ r α G, is just thecrossed product corresponding to the subgroup { e } , A ⋊ { e } α G. By the induction instages [10, Theorem 5.9], for every inclusion H (cid:31) (cid:127) / / K of subgroups of G we have a Date : September 3, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Fell bundles, induction, amenability, absorption principle. quotient map A ⋊ Kα G q KH / / / / A ⋊ Hα G ; where by a quotient map we mean a morphismof *-algebras which is surjective as a function.Another way of presenting the construction above is by considering the α − semidirectproduct B of A and G [7, VIII 4.2]. This new object is our prototypical example ofa saturated Fell bundle (see [7], where Fell bundles are called C*-algebraic bundles).The crossed product A ⋊ α G is just the cross sectional C*-algebra C ∗ ( B ) . If one nowtakes a closed subgroup H ⊂ G, then the α | H − semidirect product bundle of A and H is the reduction of B to H, B H , and the induction of covariant pairs from ( A, H, α | H )to ( A, G, α ) is Fell’s induction of B H − positive representations [7, XI 9]. In fact, every*-representation of B H is B− positive because B is saturated [7, XI 11.10]. Hence, any*-representation of B H is inducible to B . In [6] Exel and Ng construct, out of a Fell bundle B = { B t } t ∈ G , a full right B e − Hilbertmodule L e ( B ) and a *-representation Λ B : B → B ( L e ( B )) they call the regular repre-sentation of B . The reduced cross sectional C*-algebra C ∗ r ( B ) is defined as the imageof C ∗ ( B ) under the integrated form e Λ B : C ∗ ( B ) → B ( L e ( B )) of Λ B . Quite interest-ingly, all the *-representations of B e ≡ B { e } are B− positive by [7, XI 8.9]. Thenone can induce any non degenerate *-representation π : B e → B ( Y ) via Fell’s concreteinduction process [7, XI 9.24] to produce the (concretely) induced *-representationInd B e ↑B ( π ) : B → B ( Z ) . The *-representation abstractly induced by π is (by construc-tion [7, XI 9.25]) Λ B ⊗ π B → B ( L ( B ) ⊗ π Y ) , b Λ B b ⊗ π , and it is unitary equivalent to Ind B e ↑B ( π ) by [7, XI 9.26]. One may then think of C ∗ r ( B )as the universal C*-algebra for the (integrated forms of) *-representations induced fromthe trivial subgroup { e } ; and write C ∗{ e } ( B ) , C ∗ ( B { e } ) and L { e } ( B ) instead of C ∗ r ( B ) ,B e and L e ( B ) (respectively).Given a saturated Fell bundle B = { B t } t ∈ G and a closed subgroup H ⊂ G, allthe *-representations of B H are B− positive and, after some time spent consulting [7]and [6], one can produce a right C ∗ ( B H ) − Hilbert module L H ( B ) and a *-representationΛ H B : B → B ( L H ( B )) in such a way that given a *-representation T : B H → B ( Y )with integrated form e T : C ∗ ( B ) → B ( Y ) , the *-representation abstractly induced by T becomes Λ H B ⊗ e T B → B ( L H ( B ) ⊗ e T Y ) , b Λ H B b ⊗ e T . It is then natural to natural to define the H − cross sectional C*-algebra of B , C ∗ H ( B ) , as the image of C ∗ ( B ) under the integrated form of Λ H B : C ∗ ( B ) → B ( L H ( B )) . As a particular case of the construction above one may consider the trivial bundleover G with constant fibre C , T G . Then, for every closed subgroup H ⊂ G, the identities C ∗ ( G ) = C ∗ ( T G ) C ∗ r ( G ) = C ∗ r ( T G ) ( T G ) H = T H hold either by definition or by construction. It is natural to define C ∗ H ( G ) := C ∗ H ( T G )and to say G is H − amenable if the canonical quotient map q GH : C ∗ ( G ) → C ∗ H ( G )is a C*-isomorphism. The induction in stages [10, Theorem 5.9] implies that G is H − amenable for every closed subgroup H of G if and only if it is { e }− amenable (i.e.amenable in the usual sense).A Fell bundle B = { B t } t ∈ G is amenable (in Exel-Ng’s sense [6]) if the natural quotientmap q B ≡ e Λ B : C ∗ ( B ) → C ∗ r ( B ) is faithful; which is the case if G is amenable. Onemay then say B is amenable with respect to the closed subgroup H of G (or just H − amenable) if the natural quotient map q B H : C ∗ ( B ) → C ∗ H ( B ) is a C*-isomorphisms. NDUCTION, ABSORPTION AND AMENABILITY 3
Fell’s induction in stages [7, XI 12.15] then implies that a saturated Fell bundle B = { B t } t ∈ G is H − amenable for every closed subgroup H ⊂ G if and only if it is amenable.Exel’s version of Fell Absorption Principle (see [5, Section 18] and [6, Corollary2.15]) states that given a Fell bundle B = { B t } t ∈ G , a non degenerate *-representation T : B → B ( Y ) and letting lt : G → B ( L ( G )) be the left regular representation; thenthe integrated form of the *-representation T ⊗ lt : G → B ( Y ⊗ L ( G )) , ( b ∈ B t ) T b ⊗ lt t is weakly contained in the integrated form of Λ B ⊗ π B e ↑B ( π ) for some faithfuland non degenerate *-representation π : B e → B ( Z ) (one may replace “some” with “all”here). When specialized to B = T G this is Fell’s original statement.It is interesting to note that if κ : { e } → C is the trivial representation, then lt =Ind { e }↑ G ( κ ) and L ( G ) = L e ( G ) ⊗ κ C . One may then replace { e } with any closedsubgroup H ⊂ G and κ with any *-representation V of H in the paragraph above andask if the resulting statement holds. Let’s be more precise about this. Take a Fellbundle B over G, a closed subgroup H ⊂ G, a *-representation T : B → B ( Y ) anda unitary *-representation V : H → B ( Z ) . Let Ind GH ( V ) : G → B ( W ) be the unitaryrepresentation induced by V and T ⊗ Ind GH ( V ) : B → B ( Z ⊗ W ) the *-representationmapping b ∈ B t to T b ⊗ Ind VH ( G ) t . The question is whether or not T ⊗ Ind GH ( V ) isweakly contained in a set of *-representations induced from B H . To give a partial answer we consider the regular B ( W ) − projection-valued Borel mea-sure on G/H induced by V, which we denote P and is part of a system of imprimitivityfor G (see [7, XI 10 & XI 14.3]). Then h T ⊗ Ind GH ( V ) , ⊗ P i is a system of imprimitivityfor B and, in case B is saturated, we may use Fell’s Imprimitivity Theorem [7, XI 14.18]to deduce T ⊗ Ind GH ( V ) is induced from a *-representation of B H . Thus a general formof Exel’s Absoption Principle holds for saturated Fell bundles.Things get much more complicated without the saturation hypothesis, mainly be-cause one has nothing like [7, XI 14.18] in this situation; the closest result being [7, XI14.17], which is not very helpful as the example given in [7, XI 14.24] shows. We thinkthe solution to this problem is hiding in the constructions used to prove [6, Corollary2.15] and it is our intention to reveal it. Once this is solved, one may proceed asin [5] to prove C ∗ H ( B ) is (isomorphic to) a C*-subalgebra of B ( C ∗ ( B ) ⊗ C ∗ H ( G )) (weuse minimal tensor products here). The inclusion should be given in such a way thatidentifying C ∗ H ( G ) = C ∗ ( G ) via the canonical quotient map, then C ∗ ( B ) = C ∗ H ( B ) in B ( C ∗ ( B ) ⊗ C ∗ ( G )) . We will not be able to do this in full generality, but only when H is normal in G. The outline of this article is as follows. We start with a section in which we introducemost of our notation and recall Fell’s definition of positivity of *-representations withrespect to a Banach *-algebraic bundle. The main result of this section states thatif B = { B t } t ∈ G is a Fell bundle (saturated or not) and H ⊂ G a closed subgroup,then any *-representation of B H is B− positive. This answers a question raised by Fellin [7, XI 11.10] and implies any *-representation of B H can be induced to B . As alreadynoticed by Fell, the affirmative answer gives a characterization of B− positivity whichis much easier to work with than the original one and (in short) implies there is no lossin generality if one only considers Fell bundles instead of Banach *-algebraic bundleswhen developing a theory of induction of *-representations. We will go that far only incase we need it, mainly because certain grade of generality will be useful in Section 4.In Section 2 we construct the Hilbert modules L H ( B ) using Fell’s abstract inductionprocess and the theory of Hilbert modules (as presented in [8]). This way of presenting DAMIÁN FERRARO the induction process uses several key facts of [7]. We then prove two results we willrefer to as
FExell’s Absoption Principle , these being the main tools we will use in therest of the article. The last part of Section 2 is dedicated to prove *-representation(on Hilbert modules) of Banach *-algebraic modules can be induced in stages, whichis just an adaptation of [7, XI 12.15].The third section of the article is dedicated to present and solve some amenabilityquestions. For example, we will prove that if B = { B t } t ∈ G is a Fell bundle and is H ⊂ G a closed normal subgroup, then B is H − amenable if G is so. Moreover, we will provethat given a closed normal subgroup K such that H ⊂ K ⊂ G, there exists a canonicalquotient map q B KH : C ∗ K ( B ) → C ∗ H ( B ) and that this map is a C*-isomorphism when-ever it’s analogue q GKH : C ∗ K ( G ) → C ∗ H ( G ) is a C*-isomorphism. FExell’s AbsoptionPrinciple will be of key importance in this section.In the final part of this article we prove the Fell bundle version of the well knownfact that given a closed normal subgroup N of G, G is amenable if and only if both N and G/N are amenable. If now one is given a Fell bundle B over G, then the rôles of G, N and
G/N are played by B , B N and the bundle C*-completion of the partial cross-sectional bundle over G/N derived from B [7, VIII 6 & 16]. We will also construct otherC*-completions of this last partial cross-sectional bundle and relate it’s reduced crosssectional C*-algebras to the C*-algebras C ∗ H ( B ) for closed normal subgroups H ⊂ N. An addenda to Fell’s book on Banach *-algebraic bundles
Almost all the definitions in this work are taken or adapted from [7]. To start withwe will use the definitions of Banach *-algebraic bundles, C*-algebraic bundles (whichwe will call Fell bundles) and *-representations of Banach *-algebraic bundles. Werecall also the definition of strong approximate unit [7, VIII 2.11] and the fact thatevery Fell bundle has one [7, VIII 16.3].When we say B = { B t } t ∈ G is a Banach *-algebraic bundle we will be implicitlyassuming G is LCH, and when we say H ⊂ G is a subgroup we will actually mean H is a closed subgroup of the LCH group G. The unit of any group will be denoted e (or0 if the group is abelian). Integration with respect to a left invariant Haar measure on G will be represented by dt or d G t and the modular function of G will be denoted ∆ or∆ G . The letter B will also represent the disjoint union of the family of fibers { B t } t ∈ G . Recall that B itself is a topological space and that the topology of B t relative to B isthe norm topology of B t . Given a Banach *-algebraic bundle B = { B t } t ∈ G and an open or closed subset C ⊂ G, the reduction B C := { B t } t ∈ C is a Banach *-algebraic bundle with the topology andvector space operations inherited from B . If C is a subgroup (and B a Fell bundle),then B H is a Banach *-algebraic bundle (a Fell bundle, respectively).The set of continuous cross sections (with compact support) of B will be denoted C ( B ) (respectively, C c ( B )). The L − cross sectional *-algebra of B will be denoted L ( B ) , as in [7, VIII 5]. This algebra is the completion of the normed *-algebra C c ( B )equipped with the norm k k , the convolution product ( f, g ) f ∗ g and the involution f f ∗ ; which are determined by k f k = Z G k f ( t ) k dt f ∗ g ( t ) = Z G f ( r ) g ( r − t ) dr f ∗ ( t ) = ∆( t ) − f ( t ) ∗ . “FExell” is a fusion of “Fell” and “Exel”. NDUCTION, ABSORPTION AND AMENABILITY 5
The (universal) cross sectional C*-algebra of B , denoted C ∗ ( B ) and defined in [7,VIII 17.2], is the enveloping C*-algebra of L ( B ) and the canonical morphism of *-algebras from L ( B ) to C ∗ ( B ) will be denoted χ B : L ( B ) → C ∗ ( B ) . In case B is a Fellbundle, χ B is injective by [7, VIII 16.4] and we view L ( B ) as a dense *-subalgebra of C ∗ ( B ) . By [7, VIII 5.11] the existence of a strong approximate unit of B guaranteesthe existence of an approximate unit of L ( B ) . Given a Banach *-algebraic bundle B = { B t } t ∈ G and a subgroup H ⊂ G, the gen-eralized restriction map p : C c ( B ) → C c ( B H ) is defined in [7, XI 8.4] and it is givenby(1.1) p ( f )( t ) = ∆ G ( t ) / ∆ H ( t ) − / f ( t ) , ∀ f ∈ C c ( B ) , t ∈ H. A *-representation T : B H → B ( Y ) is B− positive in the sense of [7, XI 8] if forall f ∈ C c ( B ) it follows that e T p ( f ∗ ∗ f ) ≥ B ( Y ); with e T : L ( B ) → B ( Y ) beingthe integrated form of T [7, VIII 11]. The C*-integrated form of T is the unique *-representation χ B T : C ∗ ( B ) → B ( Y ) such that χ B T ◦ χ B = e T .
In case B is a Fell bundlewe think of χ B T as an extension of e T and just write e T instead of χ B T . Notation 1.2.
In the paragraph above it is implicit that Y is a Hilbert space andthat we denote B ( Y ) the set of bounded linear operators on Y. In forthcoming sectionswe will need to use *-representations of Banach *-algebraic bundles by adjointableoperators on (right) Hilbert modules; so it will be convenient to think of Hilbert spacesas right Hilbert modules over the complex field C . Hence, our inner products (eventhose of Hilbert spaces) are assumed to be linear in the second variable and the formulasfrom [7] should be modified accordingly. When we say Y A − is a Hilbert module we willbe meaning that A is a C*-algebra and that Y A is a right A − Hilbert module. TheC*-algebra of adjointable operators on Y A will be denoted B ( Y A ) . The goal of the following result is to recall (part of) Fell’s characterization of posi-tivity [7, XI 8.9] and to answer affirmatively a question raised by him in [7, XI 11.10].
Theorem 1.3.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H ⊂ G a subgroupand T : B H → B ( Y ) a *-representation. Then the following are equivalent:(1) T is B− positive.(2) For every coset α ∈ G/H, every positive integer n, all b , . . . , b n ∈ B α , and all ξ , . . . , ξ n ∈ Y, P ni,j =1 h ξ i , T b ∗ i b j ξ j i ≥ . (3) The restriction T | B e : B e → B ( Y ) is B− positive, that is to say h ξ, T b ∗ b ξ i ≥ forall b ∈ B and ξ ∈ Y. Besides, the three conditions above hold if B is a Fell bundle.Proof. The equivalence between (1) and (2) is part of the content of [7, XI 8.9] and(2) clearly implies (3). Note also that if B is a Fell bundle then claim (3) does holdbecause b ∗ b ≥ B e for all b ∈ B and the restriction T | B e is a *-representation of B e . As indicated by Fell at the end of [7, XI 11.10], to prove our statement it is enough toshow that (2) does hold under the (only) assumption of B being a Fell bundle.Assume B is a Fell bundle, take a coset α ∈ G/H, b , . . . , b n ∈ B α and ξ , . . . , ξ n ∈ Y. Let t , . . . , t n ∈ α be such that b j ∈ B t j (for every j = 1 , . . . , n, ) and set t := ( t , . . . , t n ) . Define the matrix space M t ( B ) := { ( M i,j ) ni,j =1 : M i,j ∈ B t − i t j , ∀ i, j = 1 , . . . , n } as in [4, Lemma 2.8]. Then M t ( B ) is a C*-algebra with usual matrix multiplication asproduct and ∗− transpose as involution. DAMIÁN FERRARO
The matrix M := ( b ∗ i b j ) ni,j =1 belongs to M t ( B ) and, regarding B as a B−B− equivalencebundle, it follows from the proof of [4, Lemma 2.8] that M is positive in M t ( B ) . If N ∈ M t ( B ) is the positive square root of M, then all the entries of N belong to B H and n X i,j =1 h ξ i , T b ∗ i b j ξ j i = n X i,j,k =1 h ξ i , T N k,i ∗ N k,j ξ j i = n X k =1 h n X i =1 T N k,i ξ i , n X j =1 T N k,j ξ j i ≥ T is B− positive. (cid:3) In [7, VIII 16] Fell starts with a Banach *-algebraic bundle B = { B t } t ∈ G and con-structs a Fell bundle C = { C t } t ∈ G , which he calls the bundle C*-completion of B , with equivalent representations theories. To put this in precise terms we introduce thefollowing. Definition 1.4.
We say { ρ t } t ∈ G : { B t } t ∈ G → { C t } t ∈ G is a morphism of Banach *-algebraic bundles if(1) B ≡ { B t } t ∈ G and C ≡ { C t } t ∈ G are Banach *-algebraic bundles.(2) ρ t : B t → C t is a linear function, for all t ∈ G. (3) The map ρ : B → C , sending b ∈ B t to ρ t ( b ) , is continuous.(4) There exists M ≥ k ρ ( b ) k ≤ M k b k for all b ∈ B . (5) For all b, c ∈ B it follows that ρ ( bc ) = ρ ( b ) ρ ( c ) and ρ ( b ∗ ) = ρ ( b ) ∗ . We call the M of (4) an upper bound for ρ and denote k ρ k the least upper bound of ρ. The composition of ρ ≡ { ρ t } t ∈ G with π ≡ { π t } t ∈ G : { C t } t ∈ G → { D t } t ∈ G is defined by π ◦ ρ := { π t ◦ ρ t } t ∈ G : { B t } t ∈ G → { D t } t ∈ G . The integrated form of the morphism ρ : B → C is the unique morphism of Banach*-algebras e ρ : L ( B ) → L ( C ) such that e ρ ( f ) = ρ ◦ f for all f ∈ C c ( B ) . If T : C → B ( Y )is a *-representation, then so it is T ◦ ρ : B → B ( Y ) and ( T ◦ ρ ) e = e T ◦ e ρ. The C*-integrated form of ρ, χ ρ : C ∗ ( B ) → C ∗ ( C ) , is the unique morphism of *-algebras suchthat χ ρ ◦ χ B = χ C ◦ e ρ. In this article we think of Banach *-algebras as Banach *-algebraic bundles overthe trivial group { e } . Hence, by the Definition above, morphism between Banach *-algebras are continuous. Recall that if ρ : B → C is a morphism of *-algebras with B a Banach *-algebra and C a C*-algebra, then ρ is contractive and so a morphism ofBanach *-algebras. In case the bundle C of Definition 1.4 is a Fell bundle it followsthat k ρ k ≤ ρ e : B e → C e is contractive and for all b ∈ B we have k ρ ( b ) k = k ρ ( b ∗ b ) k / ≤ k b ∗ b k / ≤ k b k . Fell constructs the C*-completion C = { C t } t ∈ G of B = { B t } t ∈ G by considering anorm k k c on the fibers of B . He defines C t as the norm completion of the quotient of B t by { b ∈ B t : k b k c = 0 } . This gives a canonical morphism ρ : B → C which we willrefer to as the canonical morphism; this construction motivates Definition 1.5 below(which will be used intensively in Section 4).By a C*-completion of a *-algebra C we mean a morphism of *-algebras ι : B → C such that C is a C*-algebra and ι ( B ) is dense in C. If there is no need to specify ι we just say C is a C*-completion of B. A *-algebra B is called reduced if it has aC*-completion ι : B → A such that ι is faithful (i.e. ι ( b ) = 0 implies b = 0). This isequivalent to say B has a faithful *-representation as bounded operators on a Hilbertspace. We now want to extend these ideas to the realm of Banach *-algebraic bundle. Definition 1.5 (c.f. [7, XI 12.6]) . We say ρ : B → C is a C*-completion (of B = { B t } t ∈ G )if it is a morphism of Banach *-algebraic bundles, C is a Fell bundle and ρ ( B t ) is densein C t for all t ∈ G. Given another C*-completion of B , κ : B → D , we say ρ : ι → κ NDUCTION, ABSORPTION AND AMENABILITY 7 is a morphism if ρ : C → D is a morphism of Banach *-algebraic bundles such that ρ ◦ ι = κ. The composition of morphism is the composition of functions (and thisdefines isomorphisms).
Remark . Given a C*-completion ρ : B → C , the set of sections e ρ ( C c ( B )) is pointwisedense in the sense that { f ( t ) : f ∈ e ρ ( C c ( B )) } is dense in C t , for all t ∈ G. Besides, C ( G )Γ ⊂ Γ and by [7, II 14.6] these properties imply e ρ ( C c ( B )) is dense in C c ( C ) in theinductive limit topology.We now present a Banach *-algebraic bundle whose unit fibre is a C*-algebra butit’s only C*-completion is the zero (or null) one. Example . Let B = C × Z be the trivial bundle with constant fiber C over theadditive group Z = { , } . Define the involution by ( λ, r ) ∗ := ( λ, r ) and the product( λ, r )( µ, s ) = ( λµ, r + s ) if r = 1 or s = 1 , ( − λµ,
0) if r = s = 1 . If ρ : B → C is a C*-completion, then either ρ : B e → C is a *-isomorphism or either C = { } . The first case is excluded because ( − ,
0) = (1 , ∗ (1 ,
1) is negative in B , so we must have C = { } and this forces C = { } . It is interesting to notice that thenorm of B satisfies k b ∗ b k = k b k , but B is not a Fell bundle.The construction of the bundle C*-completion C of the Banach *-algebraic bundle B is performed in such a way that if ρ : B → C is the canonical morphism, then for any*-representation T : B → B ( Y ) there exists a unique *-representation T ρ : C → B ( Y )such that T ρ ◦ ρ = T. We now extend this property to B− positive *-representations ofreductions of B to subgroups. Proposition 1.8.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H ⊂ G a subgroup, C the bundle C*-completion of B and ρ : B → C the canonical morphism. Then forevery B− positive *-representation T : B H → B ( Y ) there exists a unique *-representation T ρ : C H → B ( Y ) such that T ρ ◦ ρ = T. Reciprocally, for every *-representation S : C H → B ( Z ) the composition T := S ◦ ( ρ | B H ) : B H → B ( Z ) is a B− positive *-representationand T ρ = S. Proof.
Let T : B H → B ( Y ) be a B− positive *-representation and denote Y T the es-sential space of T, i.e. the closed linear span of { T b ξ : b ∈ B , ξ ∈ Y } . Then the *-representation T ′ : B H → B ( Y T ) given by T ′ b ξ = T b ξ is non degenerate and k T ′ b k = k T b k for all b ∈ B . By [7, XI 11.3] there exists a *-representation T ′′ : B → B ( W ) such that k T ′ b k ≤ k T ′′ b k for all b ∈ B e . Thus the construction above and the construction of thebundle C*-completion of B imply that for all b ∈ B , k T b k = k T ′ b ∗ b k / ≤ k T ′′ b ∗ b k / = k T ′′ b k = k T ′′ ρρ ( b ) k ≤ ρ ( b ) . Hence, the construction of C implies the existence of a unique*-representation T ρ : C H → B ( Y ) such that T ρρ ( b ) = T b for all b ∈ B H . We suggest toconsult [7, VIII 16] to see how Fell shows this last claim when G = H. Uniqueness of S follows from property (1) of ρ : B → C . Now take a *-representation S : C H → B ( Z ) . Then S c ≥ c ∈ C e and,since ρ ( b ∗ b ) = ρ ( b ) ∗ ρ ( b ) ∈ C + e for all b ∈ B , we get that S ◦ ρ ( b ∗ b ) ≥ b ∈ B . This implies T : B H → B ( Z ) , b S ρ ( b ) , is B− positive and T b = S ρ ( b ) = T ρρ ( b ) for all b ∈ B H . Thus property (1) implies T ρ = S. (cid:3) DAMIÁN FERRARO Induction of *-representations
Proposition 1.8 reveals that if one is only interested in studding the representationtheory of a Banach *-algebraic bundle, then one is allowed to work with the bundle C*-completion instead. One can do so even when working with induced representations [7,XI 12.6]. But the theory works nicely enough if we only assume the existence of strongapproximate units. Thus from now on we adopt the following conventions ,which we will refer to as “conventions (C)” or just “(C)”:(C1) By a group we mean a LCH topological group and by a (normal) subgroup wemean a (normal and) closed subgroup.(C2) When we say “ B is a Banach *-algebraic bundle” we actually mean that “ B is aBanach *-algebraic bundle over a (LCH topological) group and B has a strongapproximate unit”.Any other hypothesis will be stated explicitly.2.1. Integration and disintegration of *-representations.
We now adapt Fell’sintegration and disintegration theory [7, VIII] to representations on Hilbert modules.
Definition 2.1.
A *-representation of the Banach *-algebraic bundle B on the right A − Hilbert module Y A is a function T : B → B ( Y A ) which is linear when restrictedto any fibre; is multiplicative ( T ab = T a T b ); preserves the involution ( T a ∗ = T a ∗ ) and,for all ξ, η ∈ Y A , the function B →
A, b
7→ h T b ξ, η i , is continuous. We say T is nondegenerate if the essential space of T, defined as Y TA := span { T b ξ : b ∈ B , ξ ∈ Y A } , equals Y A . A vector ξ is cyclic for T if Y A = span { T b ξ : b ∈ B} . By Cohen-Hewitt’s Theorem [7, V 9] and (C) we have Y TA = { T b ξ : b ∈ B e , ξ ∈ Y A } . The essential space is then a closed A − submodule of Y A . By the essential part of T we mean the *-representation T ′ : B → B ( Y TA ) such that T ′ b ξ = T b ξ for all b ∈ B and ξ ∈ Y A . Note T ′ is non degenerate. Notation 2.2.
When we say T : B → B ( Y ) is a *-representation we will be meaningthat Y = Y C is a Hilbert space. For general *-representations on Hilbert modules wewill write T : B → B ( Y A ) . The only exception of this rule being the case when it is clearfrom the context that Y = A is a C*-algebra regarded as a right A − Hilbert module.Given any *-representation T : B → B ( Y A ) , the restriction T | B e : B e → B ( Y A ) iscontractive. Then for all b ∈ B we have k T b k = k T b ∗ b k / ≤ k b ∗ b k / ≤ k b k . Note also that given any ξ ∈ Y A we havelim a → b k T a ξ − T b ξ k = lim a → b kh T a ∗ a ξ, ξ i + h T b ∗ b ξ, ξ i − h T b ∗ a ξ, ξ i − h T a ∗ b ξ, ξ ik = 0 . Thus the function
B → Y A , b T b ξ, is continuous and given f ∈ C c ( B ) it makes senseto define a function e T f : Y A → Y A , e T f ξ := Z G T f ( t ) ξ dt. Moreover,(2.3) k e T f ξ k ≤ Z G k T f ( t ) kk ξ k dt = k f k k ξ k for every f ∈ C c ( B ) and ξ ∈ Y A . NDUCTION, ABSORPTION AND AMENABILITY 9
Proposition 2.4.
For every *-representation T : B → B ( Y A ) there exists a unique*-representation e T : L ( B ) → B ( Y A ) such that for all f ∈ C c ( B ) and η ∈ Y A , e T f ξ = R G T f ( t ) ξ dt. Moreover, the essential spaces of T and e T agree and ξ ∈ Y A is cyclic for T if and only if it is cyclic for e T .
Proof.
The comments preceding the statement imply T is a Banach representation inthe sense of [7, VIII 8.2], then it is integrable (in the sense of [7, VIII 11.2]) to arepresentation of e T of C c ( B ) by [7, VIII 11.3]. Then we can adapt the proof of [7, VIII11.4] to show that e T : C c ( B ) → B ( Y A ) is a *-representation, which is contractive by (2.3).Thus e T : C c ( B ) → B ( Y A ) admits a unique continuous extension e T : L ( B ) → B ( Y A ) , which is also a *-representation.The claim about non degeneracy can be proved as in [7, 11.10], which amounts toshow that for any b ∈ B , f ∈ C c ( B ) and ξ ∈ Y A it follows that T b ξ ∈ span { e T g ξ : g ∈ C c ( B ) } and that e T f ξ ∈ span { T c ξ : c ∈ B} , for every ξ ∈ Y A , b ∈ B and f ∈ C c ( B ) . (cid:3) Definition 2.5.
The L − integrated form of the *-representation T : B → B ( Y A ) is the*-representation e T : L ( B ) → B ( Y A ) given by the Proposition above. The C*-integratedform of T is the unique *-representation χ B T : C ∗ ( B ) → B ( Y A ) such that χ B T ◦ χ B = e T .
If it is convenient to do so, we will write T e instead of e T .
Remark . If T : B → B ( Y A ) and π : A → B ( Z C ) are *-representations and we formthe π − balanced tensor product Y A ⊗ π Z C , then there exists a unique *-representation T ⊗ π B → B ( Y A ⊗ π Z C ) such that ( T ⊗ π a ( ξ ⊗ π η ) = ( T a ξ ) ⊗ π η. The integratedform of T ⊗ π T ⊗ π e = e T ⊗ π . To disintegrate *-representations on Hilbert modules we adapt the ideas of [7, VIII13.2]. The key to do this is the following extension of [7, VI 19.11].
Proposition 2.7.
Let B be a Banach *-algebra, I a (not necessarily closed) *-idealof B, Y A a Hilbert module and π : I → B ( Y A ) a *-representation (i.e. a morphism of*-algebras). Then π is contractive with respect to the norm of I inherited from B. If π is non degenerate, that is to say Y A = span { π ( b ) ξ : b ∈ I, ξ ∈ Y A } , then it admits aunique extension π ′ to a *-representation of B. In case A = C and π is degenerate, italso admits an extension to a *-representation of B. Proof.
Given a ∈ I, ξ ∈ Y A and a state ϕ of A define p : B → C by p ( b ) = ϕ ( h π ( aba ∗ ) ξ, ξ i ) . Then p is positive in the sense of [7, VI 18] and, by [7, VI 18.14], it satisfies p ( b ∗ cb ) ≤k c k p ( b ∗ b ) for all c ∈ B and b ∈ I. Thus we obtain, for all c, b ∈ I (2.8) ϕ ( h π ( c ) π ( b ) π ( a ) ξ, π ( b ) π ( a ) ξ i ) ≤ k c k ϕ ( h π ( b ) π ( a ) ξ, π ( b ) π ( a ) ξ i ) . The closure of π ( I ) is a C*-subalgebra of B ( Y A ) , so there exists a net { b i } j ∈ J ⊂ I ofself adjoint elements such that { π ( b i ) } i ∈ I is bounded and lim i π ( c ) π ( b i ) = lim i π ( b i ) π ( c ) = π ( c ) for all c ∈ I. Putting c = aa ∗ and b = b i in (2.8) and taking limit we obtain that k aa ∗ k π ( aa ∗ ) − π ( aa ∗ ) ≥ ⇒ k aa ∗ kk π ( a ) k ≥ k π ( a ) k ⇒ k aa ∗ k / ≥ k π ( a ) k for all a ∈ I. Then π is bounded because k aa ∗ k / ≤ k a k for all a ∈ I. Assume π is non degenerate. In such a case the uniqueness of π ′ is immediate and, asin [7, VI 19.11], it’s existence is equivalent to the fact that given a ∈ B, b , . . . , b n ∈ I and ξ i , . . . , ξ n ∈ Y A it follows that(2.9) k n X j =1 π ( ab j ) ξ j k ≤ k a kk n X j =1 π ( b j ) ξ j k . To prove the identity above take a faithful and non degenerate *-representation ρ : A → B ( Z ) , form the tensor product Y ⊗ ρ Z ; and consider the *-representation π ⊗ ρ I → B ( Y ⊗ ρ Z ) such that ( π ⊗ ρ b ( ξ ⊗ ξ ) = π ( b ) ξ ⊗ ξ. Then π ⊗ ρ π ⊗ ρ ′ : B → B ( Y ⊗ ρ Z ) by [7, VI 19.11].If given a ∈ B, b , . . . , b n ∈ I, ξ i , . . . , ξ n ∈ Y A and ξ ∈ Z we set u := P nj =1 π ( b j ) ξ j and v := P nj =1 π ( ab j ) ξ j , then we have h ξ, ρ ( h v, v i A ) ξ i = h v ⊗ ξ, v ⊗ ξ i = k ( π ⊗ a )( u ⊗ ξ ) k ≤ k a k k u ⊗ ξ k ≤ k a k h ξ, ρ ( h u, u i A ) ξ i . This implies h v, v i A ≤ k a k h u, u i A and (2.9) follows (taking norms and square roots).In case A = C and π is non degenerate we may consider the essential space Y π :=span π ( I ) Y of π and consider the extension π ′ : B → B ( Y π ) ⊂ B ( Y ) of π : I → B ( Y π ) asa *-representation of B on B ( Y ) . (cid:3) Proposition 2.10.
Let B be a Banach *-algebraic bundle. Then for every non degen-erate *-representation π : L ( B ) → B ( Y A ) there exists a unique non degenerate *-repre-sentation T : B → B ( Y A ) such that π = e T .
Proof.
Follow the ideas of [7, VI 19.11] noticing that π can be extended to the boundedmultiplier algebra of L ( B ) by Proposition 2.7. (cid:3) Positive *-representations and induction.
Fell’s induction process of repre-sentations [7, XI] is intimately related to the notion of positivity of *-representations.To extend this process to *-representations on Hilbert modules one needs to choosewhether to extend Fell’s “concrete” or “abstract” induction processes (see [7, XI 9.26]).The two approaches are equivalent for *-representations on Hilbert spaces, the maindifference being the machinery required to develop each one of them. The concreteapproach uses Hilbert spaces and is appropriate when fine surgery is required. Mean-while, the abstract approach uses (the nowadays well known theory of) Hilbert modulesand induction of *-representations; this being the main reason why we have decidedto adopt the abstract approach. This being said, we will not hesitate on changing tothe concrete approach when necessary or to use Fell’s profound understanding of the(concrete) induction process.Let B = { B t } t ∈ G be a Banach *-algebraic bundle and H a subgroup of G. In order toinduce *-representations from B H to B , Fell equips C c ( B ) with a right C c ( B H ) − modulestructure and a kind of C c ( B H ) − valued conditional expectation.The action of u ∈ C c ( B H ) on f ∈ C c ( B ) (on the right) produces the element f u ∈ C c ( B ) determined by(2.11) ( f u )( t ) = Z H f ( ts ) u ( s − )∆ G ( s ) / ∆ H ( s ) − / ds, t ∈ G. The generalized restriction map p : C c ( B ) → C c ( B H ) is that of (1.1). If, for notationalconvenience, it is necessary to specify the groups G and H, we will denote p GH thegeneralized restriction map from C c ( B ) to C c ( B H ) . Remark . If H is normal in G then ∆ H is the restriction of ∆ G and p is exactlythe restriction map. In particular, if H = { e } then p ( f ) = f ( e ) . The fundamental properties of p and the action of C c ( B H ) on C c ( B ) are the following(see [7, XI 8.4]): p ( f ) ∗ = p ( f ∗ ) p ( f ) ∗ u = p ( f u ) ( f ∗ g ) u = f ∗ ( gu ) f ( u ∗ v ) = ( f u ) v (2.13) NDUCTION, ABSORPTION AND AMENABILITY 11 where the identities above hold for all f, g ∈ C c ( B ) and u, v ∈ C c ( B H ) . The following is the natural extension of Fell’s definition of positivity [7, XI 8].
Definition 2.14.
A *-representation T : B H → B ( Y A ) is B− positive if h ξ, e T p ( f ∗ ∗ f ) ξ i ≥ ξ ∈ Y A and f ∈ C c ( B ) . Two straightforward remarks are in order. Firstly, conventions (C) imply a *-representation T of B H is B− positive if and only if the essential part of T is B− positive.Secondly, if H = G then p ( f ) = f and h ξ, e T p ( f ∗ ∗ f ) ξ i = h e T f ξ, e T f ξ i ≥
0; proving thatevery *-representation of B G ≡ B is B− positive. Theorem 2.15.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H ⊂ G a subgroup, T : B H → B ( Y A ) a *-representation and π : A → B ( Z ) a *-representation. Considerthe following claims(1) T is B− positive.(2) For every coset α ∈ G/H, every positive integer n, all b , . . . , b n ∈ B α , and all ξ , . . . , ξ n ∈ Y A , P ni,j =1 h ξ i , T b ∗ i b j ξ j i ≥ . (3) The restriction T | B e : B e → B ( Y A ) is B− positive, that is to say h ξ, T b ∗ b ξ i ≥ for all b ∈ B and ξ ∈ Y A . (4) The *-representation T ⊗ π B H → B ( Y A ⊗ π Z ) of Remark 2.6 is B− positive(and, consequently, this claim can be replaced with any of the equivalent onesfor T ⊗ π given by Theorem 1.3).Then the first three are equivalent and imply the fourth, the converse holds if π isfaithful. All the claims hold if B is a Fell bundle.Proof. Let ρ : B ( Y A ) → B ( Y A ⊗ π Z ) be the *-representation such that ρ ( M )( ξ ⊗ π η ) = M ξ ⊗ π η. Then ρ ◦ T = T ⊗ π ρ ◦ e T = e T ⊗ π T ⊗ π e by Remark 2.6. Since ρ maps the positive cone of B ( Y A ) , B ( Y A ) + , into B ( Y A ⊗ π Z ) + , it follows that (1) implies(4). In case π is faithful then so it is ρ and B ( Y A ) + = ρ − ( B ( Y A ⊗ π Z ) + ) . Hence, (4)implies (1) in case π is faithful. All we need to do now is to prove claims (1) to (3) areequivalent, and to do this we may assume π is faithful.Clearly, (1) and (4) are equivalent and (2) implies (3). By Theorem 1.3, (4) isequivalent to say ρ ◦ T | B e = T ⊗ π | B e is B− positive, meaning that T | B e is B− positive(this is claim (3)) because ρ is faithful. At this point we know (1), (3) and (4) areequivalent and we only need to prove they imply (2).Assume (4) holds and take α ∈ G/H, n, b , . . . , b n ∈ B α and ξ , . . . , ξ n ∈ Y A as in(2). To prove that P ni,j =1 h ξ i , T b ∗ i b j ξ j i ≥ h η, ρ ( n X i,j =1 h ξ i , T b ∗ i b j ξ j i ) η i ≥ η ∈ Z. Given any η ∈ Z we have, by Theorem 1.3, h η, ρ ( n X i,j =1 h ξ i , T b ∗ i b j ξ j i ) η i = n X i,j =1 h ξ i ⊗ π η, ( T ⊗ π b ∗ i b j ( ξ j ⊗ π η ) i ≥ . This shows (2). (cid:3)
In the theorem below we prove the existence of a universal C*-algebra for the B− positive *-representations of B H . It will turn out to be the quotient of C ∗ ( B H )by the C*-ideal generated by the kernel of the integrated forms of all the B− positive*-representations of B H . Well, this is in fact an alternative definition of the C*-algebra C ∗ ( B + H ) we construct below. Theorem 2.16.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle and H a subgroupof G. Then there exists a C*-completion π C : L ( B H ) → C such that(1) For every f ∈ C c ( B ) , π C ( p ( f ∗ ∗ f )) ≥ . (2) Given any B− positive and non degenerate *-representation T : B H → B ( Y ) , there exists a *-representation π CT : C → B ( Y ) such that π CT ◦ π C = e T .
Moreover,(i) Given any two C*-completions π C : L ( B H ) → C and π D : L ( B H ) → D (both)satisfying conditions (1) and (2), there exists a unique C*-isomorphism Φ : C → D such that Φ ◦ π = κ. (ii) The map π CT in claim (2) is unique and, even with this addition, (2) holds evenif T is a (possibly degenerate) *-representation on a Hilbert module.(iii) A *-representation T : B H → B ( Y A ) is B− positive if and only if there exists amorphism of *-algebras π CT : C → B ( Y A ) such that π CT ◦ π C = e T . (iv) Given a non degenerate *-representation µ : C → B ( Y A ) there exists a unique B− positive *-representation T : B H → B ( Y A ) such that µ = π CT . Proof.
Let C be the bundle C*-completion of B and ρ : B → C the canonical morphism.Set C := C ∗ ( C H ) and let π C : L ( B ) → C ∗ ( C H ) the composition of the integrated formof ρ H : B H → C H , b ρ ( b ) , with the inclusion L ( C H ) ֒ → C ∗ ( C H ) . Take a B− positive (and possibly degenerate) *-representation T : B H → B ( Y A ) andlet π : A → B ( Z ) be a faithful and non degenerate *-representation. Then T ⊗ π B− positive by Theorem 2.15 and, by Proposition 1.8, it follows that k T b k = k ( T ⊗ π b k ≤ k ρ ( b ) k for all b ∈ B H . By repeating the proof of Proposition 1.8 we can produce a non degen-erate *-representation T ′ : C H → B ( Y A ) such that T ′ ◦ ρ H = T. If π CT : C ∗ ( C H ) → B ( Y )is the C*-integrated form of T ′ , then the identity π CT ◦ π C = e T follows by construction.Let µ : C ∗ ( C H ) → B ( Z ) be a non degenerate *-representation. Then µ | L ( C H ) is theintegrated form of a unique *-representation S : C H → B ( Z ) . If we set T := S ◦ ρ H , then S = T ′ and it follows that π CT = µ. At this point it is convenient to denote the gener-alized restrictions maps for B and C with different letters, so we write p B H : C c ( B ) → C c ( B H ) instead of p GH , and proceed analogously with C . We have µ ( π C ( p B H ( f ∗ ∗ f ))) = e S p C H (( ρ ◦ f ) ∗ ∗ ( ρ ◦ f )) ≥ f ∈ C c ( B ) . Since µ can be arranged to be faithful, it follows that π C ( p B H ( f ∗ ∗ f )) ≥ f ∈ C c ( B ) . We have managed to produce a C*-completion π C : L ( B H ) → C for which claims (1),(2), (ii) and (iv) hold. Assume we are given another C*-completion π D : L ( B H ) → D satisfying (1) and (2) and let µ, S and T be as before, with the additional requirementof µ being faithful. Then there exists a *-representation π DT : D → B ( Z ) such that π DT ◦ π D = e T .
Note π DT ( D ) = π DT ◦ π D ( L ( B H )) = µ ( C ) , then there exists a uniquemorphism of C*-algebras Ψ : D → C such that µ ◦ Ψ = π DT . Moreover, µ (Ψ ◦ π D ( f )) = e T f = π CT ( π C ( f )) = µ ( π C ( f )) for all f ∈ L ( B H ) . Thus Ψ ◦ π D = π C and it followsthat Ψ is surjective. In order to prove that Ψ is faithful, take a non degenerate andfaithful *-representation ν : D → B ( W ) . Then ν ◦ π D is the integrated form of a uniquenon degenerate *-representation R : B H → B ( W ) , which turns out to be B− positive bycondition (1). So π CR ◦ Ψ ◦ π D = π CR ◦ π C = e R = ν ◦ π D and it follows that π CR ◦ Ψ = ν. Consequently, the condition Ψ( x ) = 0 implies ν ( x ) = 0 and this forces x = 0; provingthat Ψ is a C*-isomorphism. The C*-isomorphism of claim (i) is Ψ − . NDUCTION, ABSORPTION AND AMENABILITY 13
Assume we are given a *-representation T : B H → B ( Y A ) for which there exists a*-representation κ : C → B ( Y A ) such that κ ◦ π C = e T .
Then condition (1) implies, forall η ∈ Y A and f ∈ C c ( B ) , that h η, e T p ( f ∗ ∗ f ) η i = h η, κ ◦ π C ( p ( f ∗ ∗ f ))) η i ≥ . Hence T is B− positive. This implies (iii). (cid:3) Definition 2.17.
Given a Banach *-algebraic bundle B over G and a subgroup H of G, the B + − completion of L ( B H ) is the C*-algebra C = C ∗ ( C H ) we constructedin the proof of Theorem 2.16 and χ B + H : L ( B H ) → C ∗ ( B + H ) is just π C : L ( B H ) → C. The algebra C ∗ ( B + H ) will be called the B + − cross sectional C*-algebra of B H . The map χ B H + : C ∗ ( B H ) → C ∗ ( B + H ) is, by definition, the unique morphism of *-algebras such that χ B H + ◦ χ B H = χ B + H . If the bundle B in the definition above is a Fell bundle, then the identity C ∗ ( B + H ) = C ∗ ( B H ) holds by construction and by the fact that every Fell bundle is it’s own bundleC*-completion [7, VIII 16.10].Most of the time we will think of C ∗ ( B + H ) as an universal object and not as theconcrete C*-algebra we constructed. Consequently, any time we know χ B H + is faithfulwe will identify C ∗ ( B H ) with C ∗ ( B + H ) . Corollary 2.18.
In the conditions of the Definition above and of Theorem 2.16 thefollowing are equivalent:(1) χ B H + is faithful, and hence a C*-isomorphism, between C ∗ ( B H ) and C ∗ ( B + H ) . (2) Every *-representation of B H is B− positive.(3) Every cyclic *-representation of B H on a Hilbert space is B− positive.In particular, C ∗ ( B ) = C ∗ ( B + G ) . In case B is a Fell bundle all the equivalent conditionsabove hold.Proof. Is left to the reader. (cid:3)
To describe the induction process in terms of Hilbert modules we construct “theinducing” module for the B− positive *-representations. Let B = { B t } t ∈ G be a Banach*-algebraic bundle and H a subgroup of G. Consider the B + − completion of L ( B H ) , χ B + H : L ( B H ) → C ∗ ( B + H ) , and define the *-algebra C := χ B + H ( C c ( B H )) , which is densein C ∗ ( B + H ) . We consider C c ( B ) with the action of C c ( B H ) on the left given by (2.11) and definethe C − valued sesquilinear form(2.19) C c ( B ) × C c ( B ) → C , ( f, g ) [ f, g ] p := χ B + H ( p ( f ∗ ∗ g )) . This form is linear in the second variable, positive semidefinite and [ f, g ] ∗ p = [ g, f ] ∗ p . Given any state ϕ of C ∗ ( B + H ) the function C c ( B ) × C c ( B ) → C , ( f, g ) ϕ ([ f, g ] p ) , isa pre-inner product. It then follows that | ϕ ([ f, g ] p ) | ≤ ϕ ([ f, f ] p ) / ϕ ([ g, g ] p ) / . Note this implies [ f, g ] p = 0 for all g ∈ C c ( B ) if and only if [ f, f ] p = 0 . Define I := { f ∈ C c ( B ) : [ f, f ] p = 0 } and note the quotient space C c ( B ) /I has aunique C − valued sesquilinear form( C c ( B ) /I ) × ( C c ( B ) /I ) → C , ( f + I, g + I )
7→ h f + I, g + I i p := [ f, g ] p , which is linear in the second variable, positive definite and satisfies h f + I, g + I i ∗ p = h g + I, f + I i p . If we are given f ∈ C c ( B ) and u ∈ C c ( B H ) , then0 ≤ [ f u, f u ] p = χ B + H ( u ) ∗ [ f, f ] p χ B + H ( u ) ≤ k [ f, f ] p k χ B + H ( u ) ∗ χ B + H ( u ) . In particular f u ∈ I if either f ∈ I or χ B + H ( u ) = 0 . It then follows that there exists aunique left action of C on C c ( B ) /I ( C c ( B ) /I ) × C → C c ( B ) /I, ( f + I, c ) ( f + I ) c, such that ( f + I ) χ B + H ( u ) = f u + I. Definition 2.20.
The C ∗ ( B + H ) − Hilbert module obtained by completing C c ( B ) /I (asindicated in [8, Lemma 2.16]) with respect to the C − valued inner product describedabove will be dented L H ( B ) . It’s inner product will be denoted h , i p . Remark . The map ι : C c ( B ) → L H ( B ) , f f + I, is linear, continuous in theinductive limit topology and has dense range. Indeed, the linearity and density claimsfollow immediately by construction. To prove the continuity claim fix a compact set D ⊂ G and take f ∈ C c ( B ) with support contained in D. Let α D be the measure of D in G ; β D the measure of ( D − D ) ∩ H in H and γ D := max { ∆ H ( s ) / ∆ H ( s ) − / : s ∈ ( D − D ) ∩ H } . Then k f + I k = k χ B + H ( p ( f ∗ ∗ f )) k / ≤ k p ( f ∗ ∗ f ) k / ≤ (cid:18)Z H ∆ H ( s ) / ∆ H ( s ) − / Z G k f ( r ) kk f ( rs ) k d G r d H s (cid:19) / ≤ ( α D β D γ D ) / k f k ∞ . Thus ι is continuous in the inductive limit topology by [7, II 14.3]. Remark . If B is a Fell bundle, then the C*-completion χ B + H : L ( B H ) → C ∗ ( B + H ) ≡ C ∗ ( B H ) is faithful and the condition p ( f ∗ ∗ f ) = 0 implies R G f ( t ) ∗ f ( t ) dt = p ( f ∗ ∗ f )( e ) =0 . Thus f ∈ I if and only if [ f, f ] p = 0 . Hence, the quotient C c ( B ) /I is C c ( B ) and L H ( B )is a completion of C c ( B ) . Example . If B is a Fell bundle and H = { e } , then B H = B e and the generalizedrestriction map p : C c ( B ) → B e is just the evaluation map f f ( e ) . It then followsthat for all f, g ∈ C c ( B ) we have [ f, g ] p = ( f ∗ ∗ g )( e ) = R G f ( t ) ∗ g ( t ) dt. This is exactlythe B e − valued inner product used by Exel and Ng in [6, Lemma 2.1] to construct the B e − Hilbert module L e ( B ) , so L e ( B ) = L { e } ( B ) . Example . If B is again a Fell bundle over G and H = G, then C ∗ ( B + G ) = C ∗ ( B )and L G ( B ) is the C*-algebra C ∗ ( B ) regarded as a right C ∗ ( B ) − Hilbert module.
Remark . Our conventions (C) imply L H ( B ) is C ∗ ( B + H ) − full. To prove this takeany f ∈ C c ( B H ) . By [7, II 14.8] there exists g ∈ C c ( B ) such that p ( g ) = f. Now takethe approximate unit { h λ } λ ∈ Λ ⊂ C c ( B ) of L ( B ) constructed in [7, VIII 5.11]. Then { h λ ∗ g } λ ∈ Λ converges to g in the inductive limit topology of C c ( B ) and, since p iscontinuous with respect to this topology, the net { p ( h λ ∗ g ) } λ ∈ Λ converges to p ( g ) = f in the inductive limit topology. It thus follows that {h h ∗ λ + I, g i p } λ ∈ Λ converges to χ B + H ( f ) , implying that L H ( B ) is full.We want a *-representation ρ : L ( B ) → B ( L H ( B )) such that ρ ( f )( g + I ) = f ∗ g + I forall f, g ∈ C c ( B ) . If H = { e } and B is Fell bundle, the existence of such a *-representationis guaranteed by [6, Proposition 2.6]. If now H is the whole group G and B is still a Fellbundle, then ρ is just the natural inclusion of L ( B ) in C ∗ ( B ) ⊂ B ( C ∗ ( B )) = B ( L G ( B )) . NDUCTION, ABSORPTION AND AMENABILITY 15
One can then suspect the “disintegrated form” of ρ has something to do with thedisintegrations of *-representations of L ( B ) and may consult [7] to get the formula forthe disintegrated form of ρ, those formulas being expressions (10) and (11) in [7, VIII5.8]. So we define, for every r ∈ G, b ∈ B r and f ∈ C c ( B ); the function bf ∈ C c ( B )by ( bf )( s ) := bf ( r − s ) . Then
B × C c ( B ) → C c ( B ) , ( b, f ) bf, is a multiplicative andassociative action of B on C c ( B ) with the additional property that ( bf ) ∗ ∗ g = f ∗ ∗ ( b ∗ g ) . Moreover, for any fixed f ∈ C c ( B ) the function B → C c ( B ) , b bf, is continuous inthe inductive limit topology. Proposition 2.26.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H a subgroup of G and χ B + H : L ( B H ) → C ∗ ( B + H ) the B + − completion of L ( B H ) . Construct the C ∗ ( B + H ) − Hilbertmodule L H ( B ) as explained before. Then there exists a unique *-representation Λ H B : B → B ( L H ( B )) such that Λ H B b ( f + I ) = bf + I for all b ∈ B and f ∈ C c ( B ) . Moreover,(1) Λ H B is non degenerate, and so it is it’s integrated form.(2) The integrated form e Λ H B : L ( B ) → B ( L H ( B )) is the unique *-representation of L ( B ) such that e Λ H B f ( g + I ) = f ∗ g + I, for all f, g ∈ C c ( B ) . Proof.
Uniqueness follows immediately. To prove existence take f ∈ C c ( B ) and a state ϕ of C ∗ ( B + H ) . Let π : C ∗ ( B + H ) → B ( Y ) be the GNS construction for ϕ, with cyclic vector ξ or norm one. There exists a unique non degenerate *-representation T : B H → B ( Y )such that π ◦ χ B + H = e T , this representation is B− positive by Theorem 2.16. Thefunctional µ : B e → C given by µ ( b ) := ϕ ◦ χ B + H ◦ p ( f ∗ ∗ ( bf )) is positive because forall b ∈ B we have µ ( b ∗ b ) = h ξ, g T ϕp (( fb ) ∗ ∗ ( bf )) ξ i ≥ . Using [7, VI 18.14] we deduce that | µ ( b ∗ ab ) | ≤ k a k µ ( b ∗ b ) for all a, b ∈ B e . Hence, for all b ∈ B e , a ∈ B , f ∈ C c ( B ) andevery state ϕ of C ∗ ( B + H ) we have(2.27) 0 ≤ ϕ ◦ χ B + H ◦ p (( abf ) ∗ ∗ ( abf )) ≤ k a k ϕ ◦ χ B + H ◦ p (( bf ) ∗ ∗ ( bf )) . Let { u i } i ∈ I be a strong approximate unit of B . Then the net { f ∗ ∗ ( u ∗ i u i f ) } i ∈ I = { ( u i f ) ∗ ∗ ( u i f ) } i ∈ I converges in the inductive limit to f ∗ ∗ f. On the other hand, p : C c ( B ) → C c ( B H ) is continuous in the inductive limit topology. Thuslim i k a k ϕ ◦ χ B + H ◦ p (( f u i ) ∗ ∗ ( f u i )) = k a k ϕ ◦ χ B + H ◦ p ( f ∗ ∗ f ) . A similar argument implies thatlim λ ϕ ◦ χ B + H ◦ p (( ae λ f ) ∗ ∗ ( ae λ f )) = ϕ ◦ χ B + H ◦ p (( af ) ∗ ∗ ( af )) . Then by (2.27),(2.28) 0 ≤ h af + I, af + I i p ≤ k a k h f + I, f + I i p for all f ∈ C c ( B ) and a ∈ B . By (2.27) above, for every a ∈ B there exists a unique linear and bounded mapΛ H B a : L H ( B ) → L H ( B ) such that Λ H B a ( f + I ) = af + I. Note that h Λ H B a ( f + I ) , g + I i p = χ B + H ( p (( af ) ∗ ∗ g )) = χ B + H ( p ( f ∗ ∗ ( ag ))) = h f + I, Λ H B a ∗ ( g + I ) i p ;so it follows that Λ H B a is adjointable with adjoint Λ H B a ∗ . Define Λ H B : B → B ( L H ( B )) by a Λ H B a . It is straightforward to verify that Λ H B is multiplicative and linear on each fibre, and we showed Λ H B preserves the involution.Note that, given f, g ∈ C c ( B ) , the function a
7→ h Λ H B a ( f + I ) , g + I i p = χ B + H ( p (( af ) ∗ ∗ g ))is continuous because the function B → C c ( B H ) given by a p (( af ) ∗ ∗ g ) is continuousin the inductive limit topology. Since C c ( B ) /I is dense in L H ( B ) , it follows that Λ H B is a *-representation such that Λ H B a ( f + I ) = af + I. Given any f ∈ C c ( B ) , if { u i } i ∈ I is a strong approximate unit of B thenlim λ k Λ H B u i ( f + I ) − ( f + I ) k = lim λ k χ B + H ( p (( u i f − f ) ∗ ∗ ( u i f − f ) k = 0because { ( u i f − f ) ∗ ∗ ( u i f − f ) } i ∈ I converges to 0 in the inductive limit topology of C c ( B ) . This, together with Proposition 2.4, implies both Λ H B and it’s integrated formare non degenerate.To prove claim (2) take f, g, h ∈ C c ( B ) . The function G → C c ( B ) , t h ∗ ∗ ( g ( t ) f ) , has compact support and is continuous in the inductive limit topology. Moreover,there exists a compact set U ⊂ G such that supp( h ∗ ∗ ( g ( t ) f )) ⊂ U for all t ∈ G and the integral R G h ∗ ∗ ( g ( t ) f ) dt makes sense in C c ( B ) with respect to the inductivelimit topology. In fact the integral R G g ( t ) f dt also makes sense in this topology and R G h ∗ ∗ ( g ( t ) f ) dt = h ∗ ∗ R G g ( t ) f dt. The construction of the product of C c ( B ) performedin [7] implies R G g ( t ) f dt = g ∗ f. After this we can deduce that h h + I, e Λ H B g ( f + I ) i p = Z G h h + I, g ( t ) f + I i p dt = χ B + H (cid:18) p (cid:18)Z G h ∗ ∗ ( g ( t ) f ) dt (cid:19)(cid:19) = χ B + H ( p ( h ∗ ∗ ( g ∗ f ))) = h h + I, g ∗ f + I i p ;and the identity g Λ H B g ( f + I ) = g ∗ f + I follows for all f, g ∈ C c ( B ) . (cid:3) Definition 2.29.
Given a Banach *-algebraic bundle B over G and a subgroup H of G, the H − regular *-representations of B and L ( B ) are, respectively, the *-representationΛ H B : B → B ( L H ( B )) and the integrated form e Λ H B : L ( B ) → B ( L H ( B )) given byProposition 2.26. The H − cross sectional C*-algebra for B , C ∗ H ( B ) , is the closure of e Λ H B ( L ( B )) . We define q B H : C ∗ ( B ) → C ∗ H ( B ) as the unique morphism of *-algebrassuch that q B H ◦ χ B = e Λ H B . Remark . By construction C ∗ H ( B ) is a non degenerate C*-subalgebra of B ( L H ( B )) , thus we may regard B ( C ∗ H ( B )) as the C*-subalgebra of B ( L H ( B )) formed by those M ∈ B ( L H ( B )) such that both M C ∗ H ( B ) and C ∗ H ( B ) M are contained in C ∗ H ( B ) . It isthen clear that the image of Λ H B is contained in B ( C ∗ H ( B )) and, when convenient, wewill regard Λ H B as the non degenerate *-representation Λ H B : B → B ( C ∗ H ( B )) . In case B is a Fell bundle over G, C ∗ ( B ) := C ∗ G ( B ) is just the usual cross sectional C*-algebra of B . The reduced cross sectional C*-algebra of B , denoted C ∗ r ( B ) and definedin [6], is C ∗{ e } ( B ) . These two claims hold by Examples 2.23 and 2.24.Let’s continue working under the hypotheses of Proposition 2.26 and take a B− positive*-representation S : B H → B ( Y A ) . By Theorem 2.16 there exists a unique *-representation χ B + H S : C ∗ ( B + H ) → B ( Y A ) such that χ B + H S ◦ χ B + H = e S. We can now form the balanced tensorproduct L H ( B ) ⊗ S Y A := L H ( B ) ⊗ χ B + HS Y A . The image of the (algebraic) elementary elementary tensor ( f + I ) ⊙ ξ ∈ ( C c ( B ) /I ) ⊙ Y A (with f ∈ C c ( B ) and ξ ∈ Y A ) in L H ( B ) ⊗ S Y A will be denoted f ⊗ S ξ. The map B ( L H ( B )) → B ( L H ( B ) ⊗ S Y A ) , T T ⊗ S H B : B → B ( L H ( B )) to getthe (abstractly) induced *-representation(2.31) Ind B H ( S ) : B → B ( L H ( B ) ⊗ S Y A ) , a T a ⊗ S , NDUCTION, ABSORPTION AND AMENABILITY 17 which has integrated form(2.32) g Ind B H ( S ) : B → B ( L H ( B ) ⊗ S Y A ) , f e T f ⊗ S , Note induced representations are always non degenerate by conventions (C), andalso that Ind B H ( S ) agrees with the representation induced by the essential part of S. Remark . If Y A is a Hilbert space, a close examination of the construction of L H ( B ) ⊗ S Y A in terms of C c ( B ) and e S reveals that g Ind B H ( S ) is (in Fell’s terms [7, XI9.26]) the abstractly induced *-representation Ind L ( B H ) ↑ L ( B ) ( S ) . Now [7, XI 9.26] tellsus that Ind B H ( S ) is unitary equivalent to Fell’s concretely induced *-representationInd B H ↑B ( S ) . Remark . If ρ : A → B ( Z C ) is a *-representation, then by the associativity ofbalanced tensor products there exists a unique unitary U : L H ( B ) ⊗ T ⊗ ρ ( Y A ⊗ ρ Z C ) → ( L H ( B ) ⊗ T Y A ) ⊗ ρ Z C such that U ( f ⊗ T ⊗ ρ ( ξ ⊗ ρ η )) = ( f ⊗ T ξ ) ⊗ ρ η. By Remark 2.6, using U we get theunitary equivalence of *-representationsInd B H ( T ⊗ ρ ∼ = Ind B H ( T ) ⊗ ρ B H ( T ⊗ ρ e ∼ = g Ind B H ( T ) ⊗ ρ . Example . Let B = { B t } t ∈ G be a Fell bundle andthink of the identity map id : B e → B e as a map from B e to the multiplier algebra B ( B e ) . Let’s denote Λ e B and e Λ e B the { e }− regular *-representations of B and L ( B ) , respectively. If we take the universal *-representation of B e , ρ : B e → B ( Y ) , then the*-representation used in [7, VIII 16.4] to show L ( B ) is reduced may be regarded as asubrepresentation of g Ind B e ( ρ ) = e Λ e B ⊗ ρ . Hence e Λ e B : L ( B ) → B ( L e ( B )) is a faithful*-representation.Part of our next Theorem is expressed in terms of Fell’s induced systems of imprim-itivity [7, XI 14.3]. It may be considered as the formalization of the fact that L H ( B )is the the Hilbert module behind Fell’s induction process. Notation 2.36.
Given a group G and a subgroup H ⊂ G, the natural action of G on C ( G/H ) will be denoted σ HG . More precisely, σ HGs ( f )( tH ) = f ( s − tH ) for all f ∈ C ( G/H ) and s, t ∈ G. Theorem 2.37.
Assume B = { B t } t ∈ G is a Banach *-algebraic bundle and H ⊂ G asubgroup. Then there exists a *-representation ψ H B : C ( G/H ) → B ( L H ( B )) which isnon degenerate and(1) Λ H B b ψ H B ( f ) = ψ H B ( σ HGt ( f ))Λ H B b for all b ∈ B t , t ∈ G and f ∈ C ( G/H ) . (2) Given a B− positive *-representation S : B H → B ( Y A ) , the *-representation ψ S : C ( G/H ) → B ( L H ( B ) ⊗ S Y A ) , f ψ H B ( f ) ⊗ S , is non-degenerate and for all b ∈ B t , t ∈ G and f ∈ C ( G/H ) , Ind B H ( S ) b ψ S ( f ) = ψ S ( σ HGt ( f ))Ind B H ( S ) b . (3) Assuming Y A is a Hilbert space, S is non degenerate, using the terminologyof [7, XI 14.3 pp-1181] and the unitary equivalence mentioned in Remark 2.33; ψ S is the integrated form of the projection-valued measure P induced by S and h Ind B H ( S ) , P i is the system of Imprimitivity induced by S. Proof.
We will, as usual, denote the generalized restriction and the B + − completion of L ( B H ) by p : C c ( B ) → C c ( B H ) and χ B + H : L ( B ) → C ∗ ( B + H ) , respectively.Given f ∈ C b ( G/H ) and g ∈ C c ( B ) define f g ∈ C c ( B ) by f g ( t ) := f ( tH ) g ( t ) . If S : B H → B ( Z ) is a B− positive *-representation, then for all ξ ∈ Z we have k f k ∞ h ξ, e S p ( g ∗ ∗ g ) ξ i − h ξ, e S p (( fg ) ∗ ∗ ( fg )) ξ i = h ξ, e S p ([( k f k ∞ − f ) g ] ∗ ∗ [( k f k ∞ − f ) g ]) ≥ . It then follows that h f g + I, f g + I i p ≤ k f k ∞ h g + I, g + I i p and, consequently, for every f ∈ C b ( G/H ) there exists a unique bounded operator ψ H B ( f ) : L H ( B ) → L H ( B ) such that ψ H B ( f )( g + I ) = f g + I. The identity p (( f g ) ∗ ∗ h ) = p ( g ∗ ∗ ( f ∗ g )) holds for all f ∈ C b ( G/H ) and g, h ∈ C c ( B ) . Thus h ψ H B ( f ) u, v i p = h u, ψ H B ( f ∗ ) v i p for all u, v ∈ C c ( B ) /I and it follows that ψ H B ( f )is adjointable with adjoint ψ H B ( f ∗ ) . We leave to the reader the verification of the factthat ψ H B : C b ( G/H ) → B ( L H ( B )) is a unital *-representation. Note the representation ψ H B of the statement is in fact the restriction to C ( G/H ) of the ψ H B we have justconstructed; for that reason we use the same symbol to denote both representations.To give a direct proof of the non degeneracy of ψ H B | C ( G/H ) one may proceed asfollows. Given g ∈ C c ( B ) let f ∈ C c ( G/H ) be such that f ( tH ) = 1 if t ∈ supp( g ) . Then f g = g and ψ H B ( f )( g + I ) = g + I, implying that C c ( B ) /I is contained in theessential space of ψ H B . Take t ∈ G, b ∈ B t , f ∈ C b ( G/H ) and g ∈ C c ( B ) . Then for all r ∈ Gb ( f g )( r ) = b ( f g )( t − r ) = bf ( t − rH ) g ( t − r ) = σ HGt ( f )( rH )( bg )( r ) = ( σ HGt ( f )( bg ))( r );implying that Λ H B b ψ H B ( f )( g + I ) = ψ H B ( σ HGt ( f ))Λ H B b ( g + I ) . Hence (1) follows.Claim (2) follows at once from (1) after one recalls that Ind B H ( S ) = Λ H B ⊗ S . Toprove (3) we assume Y ≡ Y A is a Hilbert space and S is non degenerate. In sucha situation we know (by Remark 2.33) that Ind B H ( S ) if unitary equivalent to Fell’sconcretely induced representation Ind B H ↑B ( S ) . We then need to specify the unitaryequivalence mentioned in Remark 2.33.We start (exactly) as in [7, XI 9], so we fix a continuous and everywhere positive H − rho function ρ on G (see [7, III 14.5]) and denote ρ the regular Borel Measure on G/H constructed from ρ (as in [7, III 13.10]). Fell’s Hilbert space X ≡ X ( S ) is our Y. Since our inner products are linear in the second variable, we will be forced to adaptFell’s formulas to our situation.Let Y = { Y α } α ∈ G/H be the Hilbert bundle over
G/H induced by S and let L ( ρ , Y )be the corresponding cross sectional Hilbert space (see [7, XI 9.7] and [7, II 15.12]). Theunitary operator E : L H ( B ) ⊗ S Y → L ( ρ , Y ) of [7, XI 9.8] intertwines the abstract andconcrete induced representations. The identity E ( ψ S ( f )( g ⊗ S ξ )) = f E ( g ⊗ S ξ ) followsat once for all f ∈ C b ( G/H ) and every elementary tensor g ⊗ S ξ. So E intertwines theintegrated form of the P mentioned in claim (3) and ψ S . (cid:3) Theorem 2.37 motivates the following.
Definition 2.38.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle and H ⊂ G asubgroup. An integrated system of H − imprimitivity for B is a tern Ξ := ( Y A , T, ψ )such that: NDUCTION, ABSORPTION AND AMENABILITY 19 (1) T : B → B ( Y A ) and ψ : C ( G/H ) → B ( Y A ) are *-representations and the essen-tial space of ψ contains that of T. (2) For all b ∈ B t , f ∈ C ( G/H ) and t ∈ G it follows that T b ψ ( f ) = ψ ( σ HGt ( f )) T b . A system ( Y A , T, ψ ) is non degenerate if T (and hence ψ ) is non degenerate. Remark . Condition (2) implies the essential space of
T, Y TA , is invariant under theaction of C ( G/H ) by ψ. Thus there exists a unique non degenerate integrated systemof H − imprimitivity ( Y TA , T ′ , ψ ′ ) such that T ′ b ξ = T b ξ and ψ ′ ( f ) ξ = ψ ( f ) ξ for all b ∈ B ,f ∈ C ( G/H ) and ξ ∈ Y TA . We call this system the essential part of ( Y A , T, ψ ) . Fell’s approach to imprimitivity systems is through projection-valued Borel measuresof
G/H.
This tool is available if one works with Hilbert spaces because projectionsabound in B ( Y ) . But if Y A is an A − Hilbert module and B ( Y A ) is not a von Neumannalgebra, it may not be possible to speak of “projection valued measures” on B ( Y A ) andit is more convenient to consider “integrated forms of the projection valued measures”,i.e. *-representations of C ( G/H ) . This approach is justified by [7, VIII 18.7 & 18.8].One may “disintegrate” a *-representations of C ( G/H ) on B ( Y A ) by taking a faithfuland non degenerate *-representation ρ : A → B ( Z ) and considering the faithful *-representation B ( Y A ) → B ( Y A ⊗ ρ Z ) , M M ⊗ ρ . Weak containment and Fell’s absorption principle.
Given a LCH topologi-cal group G one may view the representation theory of G as the representation theory ofthe trivial Fell bundle over G with constant fibre C , which we denote T G = { C δ t } t ∈ G . If H ⊂ G is a subgroup, then ( T G ) H ≡ T H and the induction of *-representations from H to G may be regarded as the induction of *-representations from C ∗ ( H ) := C ∗ ( T H ) to C ∗ ( G ) := C ∗ ( T G ) via the C ∗ ( H ) − Hilbert module L H ( G ) := L H ( T G ) . The C*-algebra C ∗ H ( G ) is defined as C ∗ H ( T G ) ⊂ B ( L H ( G )) and the map q T G H : C ∗ ( G ) → C ∗ H ( G ) willbe denoted q GH . Theorem 2.15 implies any *-representation of H can be induced to a*-representation of G (c.f. [9, Theorem 4.4]).Given a non degenerate integrated system of H − imprimitivity for T G , ( Y A , T, ψ ) , themap U : G → B ( Y A ) , U Tt := T δ t , is a unitary representation of G such that U Tt ψ ( f ) = ψ ( σ HGt ( f )) U Tt for all t ∈ G and f ∈ C ( G/H ) . It is then natural to say a tern ( Z C , U, φ )is a system of H − imprimitivity for G if(1) Z C is a C − Hilbert module.(2) U : G → B ( Z C ) is a unitary *-representation.(3) ψ : C ( G/H ) → B ( Z C ) is a non degenerate *-representation.(4) For all t ∈ G and f ∈ C ( G/H ) , U t φ ( f ) = φ ( σ HGt ( f )) U t . Note the process ( Y A , T, ψ ) ( Y A , U T , φ ) establishes a bijective correspondencebetween the non degenerated systems of H − imprimitivity of T G and the systems of H − imprimitivity for G. Definition 2.40.
The H − regular representation of G is U HG := U Λ H T G , with Λ H T G being that of Theorem 2.37. The system of H − imprimitivity for G associated to U HG is ( L H ( G ) , U HG , ψ HG ) with ψ HG := ψ H T G . Fell’s Imprimitivity Theorem [7, XI 14.18] gives Mackey’s Imprimitivity Theorembecause T G is saturated, see also [9, Theorem 7.18] and the references therein. Westate it here for future reference. Theorem 2.41.
Let G be a LCH topological group and H a subgroup of G. A nec-essary and sufficient condition for a unitary *-representation U : G → B ( Y ) to be (unitary equivalent to) a *-representation induced by a *-representation of H is the ex-istence of a non degenerate *-representation ψ : C ( G/H ) → B ( Y ) such that U t ψ ( f ) = ψ ( σ HGt ( f )) U t for all t ∈ G and f ∈ C ( G/H ) . Let B = { B t } t ∈ G be Banach *-algebraic bundle and H ⊂ G a subgroup. In general,the reductions B tHt − may be quite different for different t. As an example of thisconsider a discrete group G with a subgroup H for which there exists t ∈ H such that H ∩ ( tHt − ) = { e } . Let B = { B t } t ∈ G be the subbundle of T G such that B t = C if t ∈ H and B t = { } if t / ∈ H. Then B H = T H but all the fibers of B tHt − are { } , except for B e = C . Note also that C ∗ ( B H ) = C ∗ ( H ) and C ∗ ( B tHt − ) = C ∗ r ( H ) , thus C ∗ H ( B ) and C ∗ tHt − ( B ) = C ∗ r ( G ) may be quite different from each other. Theorem 2.42.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H ⊂ G a subgroup,fix s ∈ G and set K := sHs − . Consider the Haar measure on
K, d K t, such that d K ( t ) = d H ( s − ts ) . Assume at least one of the following conditions hold:(1) H is normal in G. (2) B is saturated.(3) B has a unitary multiplier of order s [7, VIII 3.9].Then (2.43) k e Λ H B f k = k e Λ K B f k ∀ f ∈ L ( B ) . In particular, there exists a unique morphism of *-algebras ρ : C ∗ H ( B ) → C ∗ K ( B ) suchthat ρ ◦ e Λ H B = e Λ K B . In fact ρ is a C*-isomorphism and it is also the unique nondegenerate *-representation ρ : C ∗ H ( B ) → B ( C ∗ K ( B )) such that ρ ◦ Λ H B = Λ K B . Proof.
The statement is immediate if H is normal in G. Assume B is saturated, in whichcase we adopt Fell’s notation (and construction) on conjugated representations [7, XI16]. Let T : B H → B ( Y ) be a non degenerate B− positive *-representation such that k g Ind B H ( T ) f k = k e Λ H B f k for all f ∈ L ( B ) . By [7, XI 16.7] the conjugated *-representation s T : B K → B ( Z ) is B− positive and, by [7, XI 16.19], Ind B K ( s T ) is unitary equivalent toInd B H ( T ) . Thus, for all f ∈ L ( B ) , k e Λ H B f k = k g Ind B H ( T ) f k = k g Ind B K ( s T ) f k ≤ k e Λ K B f k . By symmetry we obtain (2.43).Now assume B has a unitary multiplier u of order s. In this situation we may pro-ceed as in [7, XI 16.16], we repeat the construction to show the reader the saturationhypothesis is not really needed. Let T : B H → B ( Y ) be a non degenerate B− positive*-representation for which the *-representation χ B + H T : C ∗ ( B + H ) → B ( T ) is faithful (recall χ B + H T ◦ χ B + H = e T ). Define u T : B K → B ( Y ) by u T b := T u ∗ bu , and note that u ∗ ( u T ) = T. Set S := u T and given any f ∈ C c ( B ) define [ f u ] ∈ C c ( B ) by [ f u ]( t ) = ∆ G ( s ) − / f ( ts − ) u. For all f, g ∈ C c ( B ) and ξ, η ∈ Y : h [ f u ] ⊗ T ξ, [ f u ] ⊗ T η i == h ξ, e T p GH ([ fu ] ∗ ∗ [ gu ]) η i = Z H Z G ∆ G ( t ) / ∆ H ( t ) − / h ξ, T [ fu ]( r ) ∗ [ gu ]( rt ) η i d G rd H t = Z H Z G ∆ G ( s ) − ∆ G ( t ) / ∆ H ( t ) − / h ξ, T u ∗ f ( rs − ) ∗ g ( rts − ) u η i d G rd H t = Z K Z G ∆ G ( t ) / ∆ K ( t ) − / h ξ, S f ( r ) ∗ g ( rt ) η i d G rd K t = h f ⊗ S ξ, g ⊗ s η i . NDUCTION, ABSORPTION AND AMENABILITY 21
Thus there exists a unique unitary operator U : L K ( B ) ⊗ S Y → L H ( B ) ⊗ T Y such that U ( f ⊗ S ξ ) = [ f u ] ⊗ T ξ. Given t ∈ G, b ∈ B t , f ∈ C c ( B ) and ξ ∈ Y we have( b [ f u ])( r ) = b [ f u ]( t − r ) = ∆ G ( s ) − / bf ( t − rs − ) u = [( bf ) u ]( r ) , for all r ∈ G. Thus, U ∗ Ind B H ( T ) b U ( f ⊗ S ξ ) = U ∗ ( b [ f u ] ⊗ T ξ ) = U ∗ ([( bf ) u ] ⊗ T ξ )= Ind B K ( S ) b ( f ⊗ S ξ );implying that U intertwines Ind B H ( T ) and Ind B K ( S ) (and their integrated forms).Our choice of T guarantees that for all f ∈ L ( B ) k e Λ H B f k = k g Ind B H ( T ) f k = k g Ind B K ( S ) f k ≤ k e Λ K B f k . Since u ∗ is a unitary multiplier of order s − and s − Ks = H, we conclude that (2.43)holds.At this point we know that any of the conditions (1), (2) or (3) imply (2.43). Since theranges of both e Λ H B and e Λ K B are dense *-subalgebras of C ∗ H ( B ) and C ∗ K ( B ) , respectively,the existence of C*-isomorphism ρ : C ∗ H ( B ) → C ∗ K ( B ) such that ρ ◦ e Λ H B = e Λ K B follows.We leave the rest of the proof to the reader. (cid:3) Our version of the Absorption Principle will be sated in terms of minimal tensorproducts of *-representations of Hilbert modules. We now briefly present some standardtensor product constructions in convenient way.Given two right Hilbert modules, Y A and Z C , the (minimal) tensor product Y A ⊗ Z C is a right A ⊗ C − Hilbert module with inner product determined by the condition h ξ ⊗ η, ζ ⊗ ν i = h ξ, ζ i ⊗ h η, ν i ;which implies k ξ ⊗ η k = k ξ kk η k . The construction of Y A ⊗ Z C may be performed insidethe tensor product between the linking algebras L ( Y A ) ⊗ L ( Z C ) , in which case the A ⊗ C − valued inner product of Y A ⊗ Z C is just the restriction of the natural innerproduct of L ( Y A ) ⊗ L ( Z C ) . The results of [3, Section 5.2] imply this construction yieldsthe minimal tensor product of Hilbert modules.The description of minimal tensor products we have just exposed implies that incase Y A = A A and Z C = C C , the tensor product A A ⊗ C C is just the C*-algebra A ⊗ C regarded as a Hilbert module, this is A A ⊗ C C = ( A ⊗ C ) A ⊗ C . If Z C = Z is a Hilbert space ( C = C ), we identify A ⊗ C with A and regard Y A ⊗ Z as an A − Hilbert module. This module is also the balanced tensor product Z ⊗ ρ Y A for the trivial representation ρ : C → B ( Y A ) , λ λ . Thus, if both A and C are thecomplex field C , Y A ⊗ Z C is just the usual tensor product of Hilbert spaces.Every T ∈ B ( Y A ) defines an adjointable map L ( T ) ∈ B ( L ( Y A )) that, in the usualmatrix representation of L ( Y A ) , is given by L ( T ) R ξ ˜ η a ! := T R T ξ ˜ η a ! . By faithfully representing both L ( Y A ) and L ( Z c ) on a Hilbert space we can prove theexistence of a unique operator L ( T ) ⊗ ∈ B ( L ( Y A ) ⊗ L ( Z C )) mapping ξ ⊗ η to L ( T ) ξ ⊗ η. The subspace Y A ⊗ Z C is invariant under both L ( T ) and L ( T ∗ ) = L ( T ) ∗ and therestriction of L ( T ) ⊗ T ⊗ ∈ B ( Y A ⊗ Z C )mapping ξ ⊗ η to T ξ ⊗ η. In this very same way one can construct, for each S ∈ B ( Z C ) , the (unique) operator 1 ⊗ S ∈ B ( Y A ⊗ Z C ) mapping ξ ⊗ η to ξ ⊗ Sη.
The composition T ⊗ S := ( T ⊗ ◦ (1 ⊗ S ) = (1 ⊗ S ) ◦ ( T ⊗
1) is the unique adjointable operator of Y A ⊗ Z C mapping ξ ⊗ η to T ξ ⊗ Sη, and it’s adjoint is T ∗ ⊗ S ∗ . Putting together several facts from [7] (with some standard tricks) and extendingthe ideas of [6] we got the theorem below. The reader should notice that if one takes H = { e } (this is Exel-Ng’s situation [6]) then one may use the trivial representation H → C , t , instead of the *-representation V we will use in the proof below. Indoing so our proof is becomes a modified version of that of Exel and Ng. Theorem 2.44 (FExell’s Absorption Principle I) . Assume B = { B t } t ∈ G is a Banach*-algebraic bundle and H ⊂ G is a subgroup. Given t ∈ G and non degenerate *-representations T : B → B ( Y A ) and V : H → B ( Z C ) define H t := tHt − and the *-representations V t : H t → B ( Z C ) , V tr := V t − rt and S t : B H t → B ( Y A ) , S tb := T b . If U := Ind GH ( V ) and W C := L H ( G ) ⊗ V Z C , then the functions T ⊗ U : B → B ( Y A ⊗ W C ) , ( b ∈ B r ) T b ⊗ U r (2.45) S t ⊗ V t : B H t → B ( Y A ⊗ Z C ) , ( b ∈ B r ) S tb ⊗ V tr (2.46) are non degenerate *-representations, S t ⊗ V t is B− positive and (2.47) k ( T ⊗ U ) e f k = sup {k g Ind B H t ( S t ⊗ V t ) f k : t ∈ G } ∀ f ∈ L ( B ) . Proof.
The verification of the facts that both T ⊗ U and S t ⊗ V t are non degenerate*-representations are left to the reader. Note the restriction ( S t ⊗ V t ) | B e = ( T ⊗ | B e is B− positive. Hence Theorem 2.15 implies S t ⊗ V t is B− positive.Define the A ⊗ C − Hilbert module Y A ⊗ G W C := ℓ ( G ) ⊗ ( Y Z ⊗ W C ) , which we regardas the direct sum of G − copies of Y A ⊗ Z C . Similarly, we define *-representation T ⊗ G U : B → B ( Y A ⊗ G W C ) , b ℓ ( G ) ⊗ ( T ⊗ U ) b , which is the composition of T ⊗ U with the unital and faithful *-representationΘ : B ( Y A ⊗ W C ) → B ( Y A ⊗ G W C ) , Θ( R ) = 1 ℓ ( G ) ⊗ R. Note that ( T ⊗ G U ) e = Θ ◦ ( T ⊗ U ) e . Thus it suffices to prove(2.48) k ( T ⊗ G U ) e f k = sup {k g Ind B H t ( S t ⊗ V t ) f k : t ∈ G } ∀ f ∈ L ( B ) . We claim there exists a unique linear map L : C c ( G, Y A ⊗ G Z C ) → Y A ⊗ G W C which is continuous in the inductive limit topology and(2.49) L ( f ⊙ [ δ t ⊗ ξ ⊗ η ]) = δ t ⊗ ( ξ ⊗ [ f ⊗ V η ])for all f ∈ C c ( G ) , ξ ∈ Y A , η ∈ Z C and t ∈ G, where: • We regard C c ( G ) as C c ( T G ) and think of L H t ( G ) as a completion of C c ( G ) . • If w ∈ Y A ⊗ G Z C , f ⊙ w ∈ C c ( G, Y A ⊗ G Z C ) is given by f ⊙ w ( r ) = f ( r ) w. Uniqueness of L follows from the fact that the functions of the form f ⊙ ( δ t ⊗ ξ ⊗ η )span a dense subset of C c ( G, Y A ⊗ G Z C ) which is dense in the inductive limit topology(see [7, II 14.6]). Take functions u, v ∈ C c ( G, Y A ⊗ G Z C ) that can be expressed aselementary tensors u = f ⊙ ( δ r ⊗ ξ ⊗ η ) v = g ⊙ ( δ s ⊗ ζ ⊗ κ ) NDUCTION, ABSORPTION AND AMENABILITY 23 as explained before. If δ r,s is the Kronecker delta (equals 1 if and only if r = s ) then(2.50) h δ r ⊗ ( ξ ⊗ [ f ⊗ V η ]) , δ s ⊗ ( ζ ⊗ [ g ⊗ V κ ]) i == δ r,s h ξ, ζ i ⊗ h f ⊗ V η, g ⊗ V κ i = δ r,s h ξ, ζ i ⊗ Z H Z G ∆ G ( t ) / ∆ H ( t ) − / f ( z ) g ( zt ) h η, V t κ i d G zd H t = Z H Z G ∆ G ( t ) / ∆ H ( t ) − / f ( z ) g ( zt ) h δ r ⊗ ξ ⊗ η, (1 ⊗ ⊗ V t )( δ s ⊗ ζ ⊗ κ ) i d G zd H t = Z H Z G ∆ G ( t ) / ∆ H ( t ) − / h u ( z ) , (1 ⊗ ⊗ V t ) v ( zt ) i d G zd H t. Fix a compact set D and denote C D ( G, Y A ⊗ G Z C ) the set formed by those f ∈ C c ( G, Y A ⊗ G Z C ) with support contained in D ; this set is in fact a Banach space withthe norm k k ∞ . Let C ⊙ D be the subspace of C D ( G, Y A ⊗ G Z C ) spanned by the functionsof the form f ⊙ ( δ t ⊗ ξ ⊗ η ) with f ∈ C D ( G ) , t ∈ G, ξ ∈ Y A and η ∈ Z C . We clearly have C c ( G ) C ⊙ D ⊂ C ⊙ D . If, for every t ∈ G, we define C ⊙ D ( t ) as the closure of { u ( t ) : u ∈ C ⊙ D } , then C ⊙ D ( t ) = Y A ⊗ G Z C if t is in the interior of D and { } otherwise. By [1, Lemma5.1], the closure of C ⊙ D in C c ( G, Y A ⊗ G Z C ) with respect to the inductive limit topologyis { f ∈ C c ( G, Y A ⊗ G Z C ) : f ( t ) ∈ C ⊙ D ( t ) ∀ t ∈ G } = C D ( G, Y A ⊗ G Z C ) . Take any u, v ∈ C D ( G, Y A ⊗ G Z C ) . By the preceding paragraph there exists sequences { u n } n ∈ N and { v n } n ∈ N in C ⊙ D converging uniformly to u and v, respectively. Then forall n ∈ N there exists a positive integer m n and (for each j = 1 , . . . , m n ) elements f n,j , g n,j ∈ C D ( G ) , r n,j , s n,j ∈ G, ξ n,j , ζ n,j ∈ Y A and ζ n,j , κ n,j ∈ Z C such that u n = m n X j =1 f n,j ⊙ ( δ r n,j ⊗ ξ n,j ⊗ η n,j ) v = m n X j =1 g n,j ⊙ ( δ s n,j ⊗ ζ n,j ⊗ κ n,j ) . Let α D be the measure of D with respect to d G s and β D that of H ∩ ( D − D ) withrespect to d H t. If γ D := sup {| ∆ G ( t ) / ∆ H ( t ) − / | : t ∈ H ∩ ( D − D ) } , then (2.50) impliesthat for all p, q ∈ N we have k m p X j =1 δ r p,j ⊗ ( ξ p,j ⊗ [ f p,j ⊗ V η p,j ]) − m q X k =1 δ s q,k ⊗ ( ζ q,k ⊗ [ g q,k ⊗ V κ q,k ]) k == k Z H Z G ∆ G ( t ) / ∆ H ( t ) − / h ( u p − v q )( z ) , (1 ⊗ ⊗ V t )( u p − v q )( zt ) i d G zd H t k≤ α D β D γ D k u p − v q k ∞ . Several conclusion arise from the inequality above:(a) If u = v and v q = u q , it follows that { P m p j =1 δ r p,j ⊗ ( ξ p,j ⊗ [ f p,j ⊗ V η p,j ]) } p ∈ N is aCauchy sequence in Y A ⊗ G W C and hence has a limit L D ( { u n } n ∈ N ) . (b) If u = v and we take limit in p and q, it follows that L D ( { u n } n ∈ N ) = L D ( { v n } n ∈ N ) . Hence L D ( { u n } n ∈ N ) depends only on u and it makes sense to define a function L D : C D ( G, Y A ⊗ G Z C ) → Y A ⊗ G W C , f L D ( f ) , that can be computed using theprocedure we have described before.(c) Taking limit in p and q we obtain k L D ( u ) − L D ( v ) k ≤ √ α D β D γ D k u − v k ∞ , so L D is continuous.(d) L D ( f ⊙ ( δ t ⊗ ξ ⊗ η )) = δ t ⊗ ξ ⊗ ( f ⊗ V η ) and L D is linear when restricted to C ⊙ D . Thus L D is linear.(e) By (2.50) and the continuity of L D , (2.51) h L D ( u ) , L D ( v ) i = Z G Z H ∆ G ( t ) / ∆ H ( t ) − / h u ( s ) , (1 ⊗ ⊗ V t ) v ( st ) i d H td G s. Clearly, if E ⊂ G is a compact containing D, L E is an extension of L D . Then thereexists a unique function L : C c ( G, Y A ⊗ G Z C ) → Y A ⊗ G W C extending all the functions L D . This extension is linear and continuous in the inductive limit topology, by [7, II14.3]. Note also that L satisfies (2.49) and so has dense range.Given t ∈ G, f ∈ C c ( B ) , ξ ∈ Y A and η ∈ Z C we define [ t, f, ξ, η ] ∈ C c ( G, Y A ⊗ G Z C )by(2.52) [ t, f, ξ, η ]( r ) = ∆ G ( t ) − / δ t ⊗ T f ( rt − ) ξ ⊗ η We claim there exists a unique linear and continuous map(2.53) I : M t ∈ G L H t ( B ) ⊗ S t ⊗ V t ( Y A ⊗ Z C ) → Y A ⊗ G W C such that for all t ∈ G, f ∈ C c ( B ) ⊂ L H t ( B ) , ξ ∈ Y A and η ∈ Z C , (2.54) I ( f ⊗ S t ⊗ V t ( ξ ⊗ η )) = L ([ t, f, ξ, η ]) . The direct summand L H t ( B ) ⊗ S t ⊗ V t ( Y A ⊗ Z C ) of L t ∈ G L H t ( B ) ⊗ S t ⊗ V t ( Y A ⊗ Z C ) isgenerated by elements f ⊗ S t ⊗ V t ( ξ ⊗ η ) , with f ∈ C c ( B ) , ξ ∈ Y A and η ∈ Z C . Note the“ t ” in the tensor product indicates the direct summand the tensor belongs to.Take vectors f ⊗ S r ⊗ V r ( ξ ⊗ η ) and g ⊗ S t ⊗ V t ( ζ ⊗ κ ) . If Ψ r ( w ) := ∆ G ( w ) / ∆ H r ( w ) − / , then by (2.51) we have h L ([ r, f, ξ, η ] , L ([ t, g, ζ , κ ]) i == Z H Z G Ψ e ( w )∆ G ( rt ) − / h δ r ⊗ T f ( zr − ) ξ ⊗ η, δ t ⊗ T g ( zwt − ) ζ ⊗ V w κ i d G zd H w = Z H Z G Ψ e ( w )∆ G ( r − ) δ r,t h T f ( zr − ) ξ ⊗ η, T g ( zwr − ) ζ ⊗ V w κ i d G zd H w = Z H Z G Ψ e ( w ) δ r,t h T f ( z ) ξ ⊗ η, T g ( zrwr − ) ζ ⊗ V w κ i d G zd H w = Z H Z G Ψ r ( rwr − ) δ r,t h ξ ⊗ η, T f ( z ) ∗ g ( zrwr − ) ζ ⊗ V rrwr − κ i d G zd H w = Z H r Z G Ψ r ( w ) δ r,t h ξ ⊗ η, T f ( z ) ∗ g ( zw ) ζ ⊗ V rw κ i d G zd H r w = δ r,t h ξ ⊗ η, ( S r ⊗ V r ) e p GHr ( f ∗ ∗ g ) ( ζ ⊗ κ ) i = h f ⊗ S r ⊗ V r ( ξ ⊗ η ) , g ⊗ S t ⊗ V t ( ζ ⊗ κ ) i . Thus there exists a linear isometry I satisfying both (2.53) and (2.54). Besides, I preserves inner products.Adapting the ideas of [6], we define ρ : G → B ( Y A ⊗ G W C ) , ρ ( t ) := lt t ⊗ Y A ⊗ W C , where lt : G → B ( ℓ ( G )) is the left regular representation, which we recall is determinedby the condition lt t ( δ s ) = δ ts . Note ρ and Θ have commuting ranges, so the range of ρ commutes with that of T ⊗ G U and ( T ⊗ G U ) e (and the respective closures, of course).Let K be the image of the map I of (2.53). We claim that G · K := span { ρ ( t ) K : t ∈ G } is dense in Y A ⊗ G W C . To prove this we define, for each t ∈ G, the function µ ( t ) : C c ( G, Y A ⊗ G Z C ) → C c ( G, Y A ⊗ G Z C ) , ( µ ( t ) f )( z ) = ( lt t ⊗ Y A ⊗ Z C ) f ( z ) . In particular, µ ( t )( f ⊙ ( δ r ⊗ ξ ⊗ η )) = f ⊙ ( δ tr ⊗ ξ ⊗ η ) . Hence, L ◦ µ ( t )( f ⊙ ( δ r ⊗ ξ ⊗ η )) = L ( f ⊙ ( δ tr ⊗ ξ ⊗ η )) = δ tr ⊗ ξ ⊗ [ f ⊗ V η ] == ρ ( t )( δ r ⊗ ξ ⊗ [ f ⊗ V η ]) = ρ ( t ) L ( f ⊙ ( δ r ⊗ ξ ⊗ η )) . NDUCTION, ABSORPTION AND AMENABILITY 25
Since both L ◦ µ ( t ) and ρ ( t ) ◦ L are linear and continuous in the inductive limit topologyand agree on a dense set, it follows that L ◦ µ ( t ) = ρ ( t ) ◦ L. Thus G · K contains theimage under L of K := span { µ ( t )[ r, f, ξ, η ] : r, t ∈ G, f ∈ C c ( B ) , ξ ∈ Y A , η ∈ Z C } ⊂ C c ( G, Y A ⊗ G Z C ) . Note C ( G ) K ⊂ K . Besides, µ ( r )[ t, f, ξ, η ]( z ) = ∆ G ( t ) − / δ rt ⊗ T f ( zt − ) ξ ⊗ η. Fixing z ∈ G and varying r, t ∈ G, ξ ∈ Y A , η ∈ Z C and f ∈ C c ( B ) , the elements weobtain on the right hand side of the displayed equation above are all those of the form δ s ⊗ T b ξ ⊗ η, for arbitrary s ∈ G, b ∈ B , ξ ∈ Y A and η ∈ Z C . Since T is non degenerate,this last type of vectors span Y A ⊗ G Z C , and we conclude (using [7, II 14.3]) that K is dense in C c ( G, Y A ⊗ G Z C ) in the inductive limit topology. Hence G · K contains thedense set L ( K ) and it follows that G · K = Y A ⊗ G W C . Our next goal is to show that defining the *-representation R := M r ∈ G Ind B H r ( S r ⊗ V r )the identity ( T ⊗ G U ) b ◦ I = I ◦ R b obtains for all b ∈ B . To prove this claim we fix r, p, q ∈ G, b ∈ B r , f ∈ C c ( B ) ,g ∈ C c ( G ) , ξ, ζ ∈ Y A and η, κ ∈ Z C . For convenience we denote u and v the tensors f ⊗ S p ⊗ V p ( ξ ⊗ η ) and g ⊙ ( δ q ⊗ ζ ⊗ κ ) , respectively. Recalling (2.51) we get h ( T ⊗ G U ) b ◦ I ( u ) , L ( v ) i == h L ([ p, f, ξ, η ]) , ( T ⊗ G U ) b ∗ ( δ q ⊗ ζ ⊗ ( g ⊗ V κ )) i = h L ([ p, f, ξ, η ]) , δ q ⊗ T b ∗ ζ ⊗ (Λ HGr − ( g ) ⊗ V κ ) i = h L ([ p, f, ξ, η ]) , L (Λ HGr − ( g ) ⊙ ( δ q ⊗ T b ∗ ζ ⊗ κ )) i = Z H Z G Ψ e ( w ) h [ p, f, ξ, η ]( z ) , δ q ⊗ T b ∗ ζ ⊗ V w κ i g ( rzw ) d G zd H w = Z H Z G Ψ e ( w )∆ G ( p ) − / h δ p ⊗ T f ( zp − ) ξ ⊗ η, δ q ⊗ T b ∗ ζ ⊗ V w κ i g ( rzw ) d G zd H w = Z H Z G Ψ e ( w )∆ G ( p ) − / δ p,q h T bf ( r − zp − ) ξ ⊗ η, ζ ⊗ V w κ i g ( zw ) d G zd H w = Z H Z G Ψ e ( w )∆ G ( p ) − / δ p,q h T ( bf )( zp − ) ξ ⊗ η, ζ ⊗ V w κ i g ( zw ) d G zd H w = h L ([ p, bf, ξ, η ]) , L ( g ⊙ ( δ q ⊗ ζ ⊗ κ )) i = h I ( bf ⊗ S q ⊗ V q ( ξ ⊗ η )) , L ( v ) i = h I ◦ R b ( u ) , L ( v ) i . Since u and v are arbitrary, L has dense range and the elements like u and v spandense subspaces of their containing spaces, by linearity and continuity we get that( T ⊗ G U ) b ◦ I = I ◦ R b ; which implies that(2.55) ( T ⊗ G U ) e f ◦ I = I ◦ e R f , ∀ f ∈ L ( B ) . Consider the *-representationsΩ T : C ∗ ( B ) → B ( Y A ⊗ G W C )Ω R : C ∗ ( B ) → B ( M r ∈ G L H r ( G ) ⊗ S r ⊗ V r ( Y A ⊗ Z C )) such that Ω T ◦ e Λ G B = ( T ⊗ G U ) e and Ω R ◦ e Λ G B = e R. Since the image of e Λ G B is densein C ∗ ( B ) , (2.55) implies Ω T ( f ) ◦ I = I ◦ Ω R ( f ) for all f ∈ C ∗ ( B ) . Besides, the image ofΩ T is the closure of that of ( T ⊗ G U ) e , which commutes with the image of ρ. Thus theimages of Ω T and ρ commute.Assume we are given f ∈ C ∗ ( B ) such that Ω R ( f ) = 0 . Then, for all t ∈ G and u ∈ L r ∈ G L H r ( G ) ⊗ S r ⊗ V r ( Y A ⊗ Z C ) , Ω T ( f ) ◦ ρ ( t ) ◦ I ( u ) = ρ ( t ) ◦ Ω T ( f ) ◦ I ( u ) = ρ ( t ) ◦ I ◦ Ω R ( f )( u ) = 0 . Recalling that I ( u ) ∈ K and that G · K spans a dense subset of Y A ⊗ G W C , we deducethat Ω T ( f ) = 0 . By thinking of the image of Ω R as the quotient of C ∗ ( B ) by the kernel of Ω R , we candefine a morphism of *-algebras Φ : Ω R ( C ∗ ( B )) → Ω T ( C ∗ ( B )) such that Ψ ◦ Ω R = Ω T . Since Ψ is contractive, it follows that k Ω R ( f ) k ≥ k Ω T ( f ) k for all f ∈ C ∗ ( B ) . Theinequality k Ω R ( f ) k ≤ k Ω T ( f ) k is trivial because I is an isometry and Ω T ( f ) ◦ I = I ◦ Ω R ( f ) . Then Φ is isometric and, consequently, a C*-isomorphism. It thus followsthat for all f ∈ L ( B ) , k ( T ⊗ G U ) f k = k Ω T ( e Λ G B f ) k = k Ω R ( e Λ G B f ) k = k e R f k = k M r ∈ G g Ind B H r ( S r ⊗ V r ) f k = sup {k g Ind B H r ( S r ⊗ V r ) f k : r ∈ G } ;showing (2.48) holds and completing the proof. (cid:3) Corollary 2.56 (FExell’s Absorption Principle II) . Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H ⊂ G a subgroup, T : B → B ( Y A ) a non degenerate *-representationand ( Z C , U, Ψ) an integrated system of H − imprimitivity for G. Assume at least one ofthe following conditions holds(1) H is normal in G. (2) B is saturated.(3) There exists a set S ⊂ G such that { tHt − : t ∈ S } = { tHt − : t ∈ G } and forevery t ∈ S there exists a unitary multiplier of B of order t. Then there exists a unique *-representation π T U : C ∗ H ( B ) → B ( Y A ⊗ Z C ) such that π T U ◦ e Λ H B = ( T ⊗ U ) e . Moreover, π T U is also the unique non degenerate *-representation ρ : C ∗ H ( B ) → B ( Y A ⊗ Z C ) such that ρ ◦ Λ H B = T ⊗ U, with ρ : B ( C ∗ H ( B )) → B ( Y A ⊗ Z C ) being the unique *-representation extending ρ. Proof.
The key point of the proof is to show that(2.57) k ( T ⊗ U ) e f k ≤ k e Λ H B f k ∀ f ∈ L ( B ) . Take faithful and non degenerate *-representations µ : A → B ( V ) and ν : C → B ( W ) . Then, S : B → B (( Y A ⊗ W C ) ⊗ µ ⊗ ν ( V ⊗ W )) , ( b ∈ B t ) ( T b ⊗ U t ) ⊗ µ ⊗ ν (1 V ⊗ W ) , is a *-representation and for all f ∈ L ( B ) it follows that k e S f k = k ( T ⊗ U ) e f k . But S isunitary equivalent to ( T ⊗ µ V ) ⊗ ( U ⊗ ν W ) . Thus, to prove (2.57), we may replace T with T ⊗ µ V and U with U ⊗ ν W . In doing so one replaces Y A and Z C with Hilbertspaces, thus to prove (2.57) we may assume Y A and Z C are Hilbert spaces to start with(and forget µ and ν ). NDUCTION, ABSORPTION AND AMENABILITY 27
By Makey’s Theorem 2.41 the system (
Z, U, ψ ) is induced from some unitary *-representation V : H → B ( W ) . Then, by Theorem 2.44 and the construction of inducedrepresentations k ( T ⊗ U ) e f k ≤ sup {k e Λ H t B f k : t ∈ G } ∀ f ∈ L ( B ) . Thus Theorem 2.42 together with any of the conditions (1), (2) or (3) gives (2.57).Clearly, condition (2.57) is no other thing that the claim of the existence of a uniquemorphism of *-algebras π T U : e Λ H B ( L ( B )) → B ( Y A ⊗ Z C ) such that: (i) π T U ◦ e Λ H B =( T ⊗ U ) e , and (ii) π T U is contractive with respect to the C*-norm inherited from C ∗ H ( B ) . We define π T U as the unique continuous extension of π T U . This extension is a nondegenerate *-representation because it’s image contains the non degenerate algebra( T ⊗ U ) e ( L ( B )) . Note π T U ◦ Λ H B : B → B ( Y A ⊗ Z C ) is a non degenerate *-representation with integratedform ( T ⊗ U ) e . Thus the uniqueness of integrated forms implies π T U ◦ Λ H B = T ⊗ U. Assume, conversely, that ρ : C ∗ H ( B ) → B ( Y A ⊗ Z C ) is a non degenerate *-representationsuch that ρ ◦ Λ H B = T ⊗ U. Then π T U ◦ e Λ H B = ( T ⊗ U ) e = ( ρ ◦ Λ H B ) e = ρ ◦ e Λ H B = ρ ◦ e Λ H B , implying that π T U and ρ agree on a dense set. Thus ρ = π T U . (cid:3) Corollary 2.58.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle, H ⊂ G a normalsubgroup, T : B → B ( Y A ) a non degenerate *-representation and V : H → B ( Z C ) aunitary *-representation. Then the *-representation ( T | B H ) ⊗ V is B− positive and thereexists a unique *-representation π T V : C ∗ H ( B ) → B ( Y A ⊗ Z C ) such that π T V ◦ e Λ H B =( T ⊗ Ind GH ( V )) e . Moreover,(1) π T V is also the unique non degenerate *-representation ρ : C ∗ H ( B ) → B ( Y A ⊗ Z C ) such that ρ ◦ Λ H B = T ⊗ Ind GH ( V ) . (2) If the integrated form of ( T | B H ) ⊗ V factors via χ B + H : L ( B H ) → C ∗ ( B + H ) througha faithful *-representation of C ∗ ( B + H ) , then π T V is faithful. To accomplish thisone may consider, for example, the trivial representation V : H → C and a*-representation T like the one we will construct before stating Corollary 3.9.Proof. The existence and uniqueness of π T V follows from FExell’s Absorption Prin-ciple II because U := Ind GH ( V ) is, by construction, part of an integrated system of H − imprimitivity. If condition (2) holds, by FExell’s Absorption Principle I we have k e Λ H B f k ≥ k π T V ( e Λ H B f ) k = sup {k g Ind B H (( T | B H ) ⊗ V t ) f k : t ∈ G } ≥≥ k g Ind B H (( T | B H ) ⊗ V ) f k = k e Λ H B f ⊗ ( T | B H ) ⊗ V k = k e Λ H B f k . It thus follows that π T V is a isometry and hence faithful. (cid:3)
Induction in stages.
In [7, XI 12.15] Fell shows that *-representations may beinduced in stages. For general *-representations on Hilbert modules this becomes theresult below.
Theorem 2.59.
Let B = { B t } t ∈ G be a Banach *-algebraic. Consider two subgroups, H and K, with H ⊂ K ⊂ G and a *-representation S : B H → B ( Y A ) . Then the followingare equivalent:(1) S is B− positive.(2) S is B K − positive and Ind B K H ( S ) is B− positive. If the conditions above hold, then there exists a unique unitary (2.60) U : L K ( B ) ⊗ Ind B KH ( S ) ( L H ( B K ) ⊗ S Y A ) → L H ( B ) ⊗ S Y A mapping f ⊗ Ind B KH ( S ) ( u ⊗ S ξ ) to f u ⊗ S ξ for all f ∈ C c ( B ) and u ∈ C c ( B K ) . Moreover,this unitary intertwines
Ind B K (Ind B K H ( S )) and Ind B H ( S ) . Proof. If Y where a Hilbert space the proof would follow at once by [7, XI 12.15]. Wewill present a complete proof instead of indicating how to modify Fell’s proof to thegeneral case, mainly because we think this is shorter and more convenient to the reader.The generalized restriction maps will be denoted indicating the groups as follows: p KH : C c ( B K ) → C c ( B H ) , p KH ( f )( t ) = ∆ K ( t ) / ∆ H ( t ) − / f ( t ) . A direct computation shows that p KH ◦ p GK = p GH . If condition (1) holds, then S | B e is B− positive and hence B K − positive. By Theo-rem 2.15 this implies S is B K − positive. Thus to prove the equivalence between (1)and (2) we can assume S is B K − positive and consider the induced *-representationInd B K H ( S ) : B K → B ( L H ( B K ) ⊗ S Y A ) . Given any f, g ∈ C c ( B ) , u, v ∈ C c ( B K ) and ξ, η ∈ Y A we have:(2.61) h u ⊗ S ξ, g Ind B K H ( S ) p GK ( f ∗ ∗ g ) ( v ⊗ S η ) i = h u ⊗ S ξ, p GK ( f ∗ ∗ g ) ∗ v ⊗ S η ) i = h ξ, e S p KH ( u ∗ ∗ p GK ( f ∗ ∗ g ) ∗ v ) ⊗ S η ) i = h ξ, e S p KH ( p GK (( fu ) ∗ ∗ ( gv ))) ⊗ S η ) i = h ξ, e S p GH (( fu ) ∗ ∗ ( gv )) ⊗ S η ) i = h f u ⊗ S ξ, gv ⊗ S η ) i . Assume S is B− positive. To show that Ind B K H ( S ) is B− positive it suffices to showthat h ζ , Ind B K H ( S ) p GK ( f ∗ ∗ f ) ζ i ≥ f ∈ C c ( B ) and every ζ ∈ L H ( B K ) ⊗ S Y A beinga finite sum of elementary tensors. Take f ∈ C c ( B ) and ζ = P nj =1 u j ⊗ S ξ j , with u , . . . , u n ∈ C c ( B K ) and ξ , . . . , ξ n ∈ Y A . Then (2.61) implies h ζ , g Ind B K H ( S ) p GK ( f ∗ ∗ f ) ζ i = h n X j =1 f u j ⊗ S ξ j , n X k =1 f u k ⊗ S ξ k i ≥ S is B− positive we take f ∈ C c ( B )and ξ ∈ Y A . By [7, VIII 5.11] and conventions (C) L ( B K ) has an approximate unit { u λ } λ ∈ Λ ⊂ C c ( B K ) such that both { u λ ∗ v } λ ∈ Λ and { v ∗ u λ } λ ∈ Λ converge to v in theinductive limit topology, for all v ∈ C c ( B K ) . Moreover, by adapting the proof of [7, VIII5.11] it can be shown that { u ∗ λ ∗ v ∗ u λ } λ ∈ Λ converges to v in the inductive limit topology.With v = p GH ( f ∗ ∗ f ) , the continuity of the generalized restrictions with respect to theinductive limit topologies and (2.61) imply h ξ, e S p GH ( f ∗ ∗ f ) ξ i = h ξ, e S p KH ( p GK ( f ∗ ∗ f )) ξ i = lim λ h ξ, e S p KH ( u ∗ λ ∗ p GK ( f ∗ ∗ f ) ∗ u λ ) ξ i = lim λ h u λ ⊗ S ξ, g Ind B K H ( S ) p GK ( f ∗ ∗ f ) ( u λ ⊗ S η ) i ≥ . Hence (2) implies (1).From now on we assume S is B− positive. In this situation (2.61) implies the existenceof a unique linear isometry U as in (2.60). In fact (2.61) can now be interpreted asthe fact that U preserves inner products. Thus U is a unitary if and only if it issurjective. To prove this is the case fix f ∈ C c ( B ) , ξ ∈ Y A and take an approximate NDUCTION, ABSORPTION AND AMENABILITY 29 unit { u λ } λ ∈ Λ ⊂ C c ( B K ) as the one we considered before. Then some straightforwardarguments show thatlim λ k f ⊗ S ξ − U ( f ⊗ Ind B KH ( S ) u λ ⊗ S ξ ) k = lim λ k ( f − f u λ ) ⊗ S ξ k = lim λ kh ξ, e S p KH ( p GK ( f ∗ ∗ f )+ u ∗ λ ∗ p GK ( f ∗ ∗ f ) ∗ u λ − u ∗ λ ∗ p GK ( f ∗ ∗ f ) − p GK ( f ∗ ∗ f ) ∗ u λ ) ξ ik = 0 . Thus U has dense range and is in fact surjective because it is an isometry.Finally, for all b ∈ B , f ∈ C c ( B ) , u ∈ C c ( B K ) and ξ ∈ Y A we have U ∗ Ind B H ( S ) b U ( f ⊗ Ind B KH ( S ) u ⊗ S ξ ) = U ∗ ( b ( f u ) ⊗ S ξ ) = U ∗ (( bf ) u ⊗ S ξ )= ( bf ) ⊗ Ind B KH ( S ) u ⊗ S ξ = Ind B K (Ind B K H ( S )) b ( f ⊗ Ind B KH ( S ) u ⊗ S ξ ) . Thus the proof follows by the density of elementary tensor products. (cid:3)
When specialized to Fell bundles the induction in stages is a statement about regular*-representations of subgroups. Recall that after Theorem 2.15 there is no need to checkpositivity of *-representations when working with Fell bundles.
Corollary 2.62.
Assume B is a Fell bundle over G and consider two subgroups of G, H and K, such that H ⊂ K ⊂ G. If Λ H B K : B K → B ( L H ( B K )) is the H − regular*-representation of B K , then there exists a unitary U : L K ( B ) ⊗ Λ H B K L H ( B K ) → L H ( B ) mapping f ⊗ Λ H B K u to f u, for all f ∈ C c ( B ) and u ∈ C c ( B K ) . Moreover, this unitaryintertwines
Ind B K (Λ H B K ) and Λ H B . Proof.
Follows directly from Theorem 2.59. (cid:3)
Corollary 2.63. If B is a Fell bundle over G and H is a subgroup of G, then the H − regular *-representation e Λ H B : L ( B ) → B ( L H ( B )) is faithful.Proof. It suffices to show that e Λ H B ⊗ e Λ e B H L ( B ) → B ( L H ( B ) ⊗ e Λ e B H L e ( B H )) is faith-ful. By construction e Λ H B ⊗ e Λ e B H ≡ Ind B H (Λ e B H ) , and Corollary 2.62 implies this *-representation is unitary equivalent to e Λ e B : L ( B ) → B ( L e ( B )); the last being faithfulby Remark 2.35. (cid:3) Amenability
Recall from Definition 2.29 that given a Banach *-algebraic bundle B over G and asubgroup H ⊂ G, the H − cross sectional C*-algebra C ∗ H ( B ) ⊂ B ( L H ( B )) is the imageof the *-representation q B H : C ∗ ( B ) → B ( L H ( B )) , this representation being the uniquemaking the diagram below a commutative one L ( B ) e Λ H B / / χ B $ $ ■■■■■■■■■ C ∗ H ( B ) C ∗ ( B ) q B H : : ✉✉✉✉✉✉✉✉✉ If B is a Fell bundle and H = { e } , we write q B e instead of q B{ e } and note C ∗ e ( B ) is Exel-Ng’s [6] reduced cross sectional C*-algebra C ∗ r ( B ); and also that q B e : C ∗ ( B ) → C ∗ r ( B ) isthe canonical quotient map. In this setting B is said to be amenable if q B e is injective. Proposition 3.1.
Let B be a Banach *-algebraic bundle over G and consider subgroups H ⊂ K ⊂ G. Then there exists a unique morphism of *-algebras q B KH making thefollowing a commutative diagram L ( B ) χ B ≡ e Λ G B ~ ~ e Λ K B (cid:15) (cid:15) e Λ H B (cid:28) (cid:28) C ∗ K ( B ) q B KH ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ C ∗ ( B ) ≡ C ∗ G ( B ) q B K ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ q B H / / C ∗ H ( B ) Moreover, q B KH is surjective and q B H is a C*-isomorphism if and only if both q B KH and q B K are C*-isomorphisms. In case B is a Fell bundle and q B K H : C ∗ ( B K ) → C ∗ H ( B K ) is aC*-isomorphism, then q B HK is a C*-isomorphism.Proof. Since q B K is surjective, q B KH is unique if it exists and to prove it’s existence itsuffices to show that k q B H ( f ) k ≤ k q B K ( f ) k for all f ∈ C ∗ ( B ) . Take a non degenerate and faithful *-representation π : C ∗ ( B + H ) → B ( Y ) . Then thereexists a *-representation T : B H → B ( Y ) such that π = χ B + H T . By Theorem 2.59, for all f ∈ L ( B K ) we have k q B H ( f ) k = k q B H ( f ) ⊗ π k ≡ k q B H ( f ) ⊗ T k = k q B K ( f ) ⊗ Ind B KH ( T ) ⊗ ⊗ T k ≤ k q B K ( f ) k . In case B is a Fell bundle and q B K H is a C*-isomorphism, Corollary 2.62 implies the*-representation q B K ⊗ q B KH C ∗ ( B ) → B ( L K ( B ) ⊗ q B KH L H ( B K )) is unitary equivalent to q B H . Then for all f ∈ C ∗ ( B ) we have k q B KH ( q B K ( f )) k = k q B H ( f ) k = k q B K ( f ) ⊗ q B KH k = k q B K ( f ) k ;and this implies q B KH is a C*-isomorphism.Note q B H , q B K and q B KH are surjective. In case q B H = q B KH ◦ q B K is a C*-isomorphism, then q B K must be faithful and hence a C*-isomorphism and this forces q B KH = q B H ◦ ( q B K ) − tobe a C*-isomorphism. (cid:3) Notation 3.2. If B = T G we will write q GH : C ∗ ( G ) → C ∗ H ( G ) and q GKH : C ∗ K ( G ) → C ∗ H ( G ) instead of q T G H : C ∗ ( T G ) → C ∗ H ( T G ) and q T G HK : C ∗ K ( T G ) → C ∗ H ( T G ) , respectively.After the Corollary above, for a Fell bundle B over G the diagram (on the left below)of inclusions of subgroups of G gives the commutative diagram of surjective morphismof C*-algebras on the right G H jJ x x ♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ? _ o o K ?(cid:31) O O { e } (cid:31) ? O O _? o o f f ◆◆◆◆◆◆◆◆◆◆◆◆◆ C ∗ ( B ) q B H / / / / q B e ( ( ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ q B K (cid:15) (cid:15) (cid:15) (cid:15) C ∗ H ( B ) q B He (cid:15) (cid:15) (cid:15) (cid:15) C ∗ K ( B ) q B KH ♠♠♠♠♠♠♠♠♠♠♠♠♠ q B Ke / / / / C ∗ r ( B )(3.3)We do not know (in general) if q B KH is a C*-isomorphism if and only if q B K H is so, butwe will be able to prove this assuming both H and K are normal in G and B is a Fellbundle. As a consequence of this we will get that if B is amenable then so it is B K . Alot of work will be needed to prove this claim.
NDUCTION, ABSORPTION AND AMENABILITY 31
We begin with a Lemma saying that the induction process followed by a restrictionincreases the norm of integrated forms, the precise form of this claim being the followingone.
Lemma 3.4 (c.f. [7, XI 11.3]) . Let B = { B t } t ∈ G be a Banach *-algebraic, H ⊂ G a subgroup and S : B H → B ( Y A ) a non degenerate B− positive *-representation. If T : B H → B ( L H ( B ) ⊗ S Y A ) is the restriction of Ind B H ( S ) : B → B ( L H ( B ) ⊗ S Y A ) , then k e S f k ≤ k e T f k ≤ k χ B + H ( f ) k for all f ∈ L ( B H ) . Proof. If ρ : A → B ( Z ) is a faithful and non degenerate *-representation, thenInd B H ( S ⊗ π
1) = Ind B H ( S ) ⊗ π T ⊗ π B H ( S ⊗ π
1) to B H . Besides, k e S f k = k ( S ⊗ π e f k and k e T f k = k ( T ⊗ π e f k . Hence, by replacing S with S ⊗ π , we mayassume Y A is a Hilbert space.Since Y ≡ Y A is a Hilbert space, we may use the concretely induced *-representationInd B H ↑B ( S ) instead of Ind B H ( S ) . This has the advantage of describing L H ( B ) ⊗ S Y as L ( ρ , Y ) (exactly as in the proof of Theorem 2.37). In [7, XI 9.9 - 9.20] Fell describesa (continuous) action τ : B × Y → Y , ( b, y ) τ b y, with the following properties:(1) For all s, t ∈ G, b ∈ B s and y ∈ Y tH ; τ b y ∈ Y stH . (2) There exists a continuous function ζ : G × G/H → R such that for all s, t ∈ G,b ∈ B s and y ∈ Y tH ; k τ b y k ≤ k b kk y k ζ ( s, tH ) . (3) For all s, t ∈ G and b ∈ B s the function B s × Y tH → Y stH , ( b, y ) τ b y, isbilinear.(4) For all f ∈ C c ( Y ) , s, t ∈ G and b ∈ B s ; [Ind B H ↑B ( S ) b f ]( tH ) = τ b g ( s − tH ) . (5) The *-representation S ′ : B H → B ( Y H ) , b τ b | Y H , is unitary equivalent to S. Fix f ∈ C c ( B H ) and g ∈ C c ( Y ) . Given ϕ ∈ C c ( G/H ) + such that R G/H ϕ dρ = 1 wehave(3.5) |h ϕg, e T f ϕg i − h g ( H ) , e S f g ( H ) i| ≤ |h ϕg, e T f ϕg i − h g ( H ) , f S ′ f g ( H ) i|≤ Z H Z G/H | ϕ ( t − x ) ϕ ( x ) h g ( x ) , τ f ( t ) ( g ( t − x )) i − ϕ ( x ) h g ( H ) , τ f ( t ) g ( H ) i| dρ ( x ) d H ( t ) ≤ Z H Z G/H ϕ ( x ) | ϕ ( t − x ) − ϕ ( x ) |k g k ∞ k f ( t ) k ζ ( t, t − x ) dρ ( x ) d H ( t )++ Z H Z G/H ϕ ( x ) |h g ( x ) , τ f ( t ) ( g ( t − x )) i − h g ( H ) , τ f ( t ) g ( H ) i| dρ ( x ) d H ( t ) . Let N be the set of compact neighbourhoods of H ∈ G/H ordered by decreasinginclusion: U ≤ V if an only if V ⊂ U. Choose, for each U ∈ N , ϕ U ∈ C c ( G/H ) + withsupport contained in U and R G/H ϕ U dρ = 1 . We claim that(3.6) lim U sup { ϕ ( x ) | ϕ ( t − x ) − ϕ ( x ) |k g k ∞ k f ( t ) k ζ ( t, t − x ) : t ∈ G, x ∈ U } = 0 . Suppose this is not the case. Then there exists a subnet { U i } i ∈ I of { U } U ∈N , ε > { t i } i ∈ I ⊂ G and { x i } i ∈ I such that: x i ∈ U i and(3.7) ϕ ( x i ) | ϕ ( t − i x i ) − ϕ ( x i ) |k g k ∞ k f ( t i ) k ζ ( t i , t − i x i ) > ε, both conditions holding for all i ∈ I. This forces { t i } i ∈ I to be contained in supp( f ) and { x i } i ∈ I to converge to H. Thus, passing to a subnet, we may assume { t i } i ∈ I converges to a point t ∈ supp( f ) . Then the left hand side of (3.7) converges to ϕ ( H ) | ϕ ( t − H ) − ϕ ( H ) |k g k ∞ k f ( t ) k ζ ( t, t − H );which is zero because t − H = H. This proves (3.6).Adapting the arguments of the paragraph above we can show thatlim U sup { ϕ ( x ) |h g ( x ) , τ f ( t ) ( g ( t − x )) i − h g ( H ) , τ f ( t ) g ( H ) i| : t ∈ G, x ∈ U } = 0 . It then follows from (3.5) thatlim U h ϕ U g, e T f ϕ U g i = h g ( H ) , e S f g ( H ) i and a similar argument implies lim U h ϕ U g, ϕ U g i = h g ( H ) , g ( H ) i . We then get that for all f ∈ C c ( B H ) and g ∈ C c ( Y ) : k e S f g ( H ) k = h g ( H ) , e S f ∗ ∗ f g ( H ) i = lim U h ϕ U g, e T f ∗ ∗ f ϕ U g i ≤ lim U k e T f k lim U h ϕ U g, ϕ U g i≤ k e T f k lim U h g ( H ) , g ( H ) i = k e T f k k g ( H ) k . This implies that k e S f k ≤ k e T f k because the fact that Y is a Banach bundle implies Y H = { g ( H ) : g ∈ C c ( B ) } . Note T | B H is B− positive because ( T | B H ) | B e = T | B e is B− positive. Then Theo-rem 2.16 implies k e T f k ≤ k π CT ( χ B + H ( f )) k ≤ k χ B + H ( f ) k for all f ∈ L ( B H ) . (cid:3) The following result implies that that given a subgroup K ⊂ G, B ( C ∗ K ( B )) containsall the C*-algebras C ∗ ( B + H ) for closed subgroups H ⊂ K. Corollary 3.8.
Let B be a Banach *-algebraic bundle over G and consider subgroups H ⊂ K ⊂ G. If S : B H → B ( L K ( B )) is the restriction of Λ K B : B → B ( L K ( B )) and χ B + H S : C ∗ ( B + H ) → B ( L K ( B )) is the unique *-representation such that χ B + H S ◦ χ B + H = e S, then χ B + H S is faithful. Moreover, if we extend the inclusion C ∗ K ( B ) ⊂ B ( L K ( B )) to aninclusion B ( C ∗ K ( B )) ⊂ B ( L K ( B )) , then χ B + H S ( C ∗ ( B + H )) ⊂ B ( C ∗ K ( B )) and we may think C ∗ ( B + H ) ⊂ B ( C ∗ K ( B )) ⊂ B ( L K ( B )) . Proof.
First of all note S is B− positive because it is the restriction of a *-representationof B , so the existence of χ B + H S is guaranteed by the universal property of C ∗ ( B + H ) . Writing S HK : B H → B ( L K ( B )) instead of S : B H → B ( L K ( B )) and using the maps q providedby Proposition 3.1 we get q B KH ◦ S KH = S HH and, in integrated forms, q B KH ◦ ] S KH = ] S HH . Here q B KH : B ( C ∗ K ( B )) → B ( C ∗ H ( B )) is the natural extension of q B KH . We then getthat q B KH ◦ χ B + H S KH = χ B + H S HH and this implies that χ B + H S KH is faithful if χ B + H S HH is so. Hence to prove this Corollary wemay assume K = H and, in doing so, write S instead of S HH . Let π : C ∗ ( B + H ) → B ( Y ) be a non degenerate and faithful *-representation. Thenthere exists a B− positive *-representation T : B H → B ( Y ) such that π ◦ χ B + H = e T , meaning that π = χ B + H T . By Theorem 2.59, S ⊗ T B H of Ind B H ( T ) . Thus for all f ∈ L ( B H ) we have k χ B + H ( f ) k = k e T f k ≤ k ( S ⊗ T e f k = k e S f k ≤ k χ B + H S ◦ χ B + H ( f ) k ≤ k χ B + H ( f ) k . NDUCTION, ABSORPTION AND AMENABILITY 33
It is then clear that χ B + H S is an isometry when restricted to the dense subalgebra χ B + H ( L ( B H )) and this implies χ B + H S is faithful. (cid:3) In the proof above we produced a *-representation R := Ind B H ( T ) of B on a Hilbertspace Z := L H ( B ) ⊗ T Y such that setting R ′ := R | B H , R ′ is B− positive and therespective *-representation χ B + H R ′ : C ∗ ( B + H ) → B ( Z ) is faithful. In case B is a Fell bundle, C ∗ ( B + H ) ≡ C ∗ ( B H ) and we regard χ B + H R ′ ≡ χ B H R ′ as the extension of f R ′ : L ( B H ) → B ( Z )to C ∗ ( B H ) . In short, if B is a Fell bundle then one may get a faithful *-representationof C ∗ ( B H ) by integrating a restriction of a *-representation of B . The following result gives an inclusion C ∗ H ( B ) ⊂ B ( C ∗ ( B ) ⊗ C ∗ H ( G )) , in fact one maytake C ∗ K ( B ) instead of C ∗ ( B ) , provided that H ⊂ K. Corollary 3.9 (c.f. [5, Propositions 18.6 & 18.7]) . Assume B be a Banach *-algebraicbundle over G, we have closed subgroups H ⊂ K ⊂ G and at least one of the conditions(1), (2) or (3) of Corollary 2.56 holds. Consider the integrated (universal) systemsof imprimitivity ( L K ( B ) , Λ K B , ψ K B ) and ( L H ( G ) , U HG , ψ HG ) for B and G, respectively.Then the *-representation Ψ B KH := π Λ K B U HG : C ∗ H ( B ) → B ( C ∗ K ( B ) ⊗ C ∗ H ( G )) provided by Corollary 2.56 is faithful.Proof. Let q B KH : C ∗ K ( B ) → C ∗ H ( B ) be the map provided by Proposition 3.1. Then q B KH ⊗ C ∗ H ( G ) ◦ Ψ B KH = Ψ B HH and it suffices to proveΨ B HH : C ∗ H ( B ) → B ( C ∗ H ( B ) ⊗ C ∗ H ( G ))is faithful.We denote V the trivial representation of G on C ( V t = 1) and V ′ the restriction of V to H. Take a non degenerate *-representation R : B → B ( Z ) as the one we constructedbefore stating the corollary we are trying to prove; we also set R ′ := R | B H . Then R ′ ⊗ V ′ is unitary equivalent to R ′ and this implies χ B + H R ′ ⊗ V ′ is unitary equivalent to χ B + H R ′ and soit is also faithful. Hence FExell’s Absorption Principle I implies, for all f ∈ L ( B ) , (3.10) k e Λ H B f k = k e Λ H B f ⊗ R ′ ⊗ V ′ k = k g Ind B H ( R ′ ⊗ V ′ ) f k ≤ k ( R ⊗ Ind GH ( V ′ )) e f k . Consider the (unique) *-representation ρ : C ∗ H ( B ) → B ( Z ) such that ρ ◦ e Λ H B = e R (recall R is induced from a B− positive *-representation of B H ). Notice that ρ is non de-generate and ρ ◦ Λ H B = R. Let µ : C ∗ H ( G ) → B ( L H ( G ) ⊗ V ′ C ) the unique *-representationsuch that µ ◦ e U HG = g Ind GH ( V ′ ); which is also the unique non degenerate one such that µ ◦ U HG = Ind GH ( V ′ ) . By construction, the non degenerate *-representation ρ ⊗ µ : C ∗ H ( B ) ⊗ C ∗ H ( G ) → B ( Z ⊗ ( L H ( G ) ⊗ V ′ B ))satisfies, for all r ∈ G and b ∈ B r ,ρ ⊗ µ ◦ Ψ B HH (Λ H B b ) = ρ ⊗ µ (Λ H B b ⊗ U HGr ) = R b ⊗ Ind GH ( V ′ ) r = ( R ⊗ Ind GH ( V ′ )) b . Then, for all f ∈ L ( B ) ,ρ ⊗ µ ◦ Ψ B HH ( e Λ H B f ) = ( R ⊗ Ind GH ( V ′ )) e f . This last identity together with (3.10) gives k e Λ H B f k ≤ k ρ ⊗ µ ◦ Ψ B HH ( e Λ H B f ) k ≤ k Ψ B HH ( e Λ H B f ) k ≤ k e Λ H B f k for all f ∈ L ( B ) . Thus Ψ B HH is an isometry and the proof is complete. (cid:3) It is important to note the defining properties of the maps Ψ B HK are the non degen-eracy and that fact thatΨ B HK (Λ H B b ) = Λ K B b ⊗ U HGs ∀ b ∈ B s , s ∈ G. Corollary 3.11.
Let B be a Banach *-algebraic bundle over G and consider subgroups H ⊂ K ⊂ G. If q GHK is injective and either (a) B is saturated or (b) both H and K arenormal, then q B HK is faithful. In particular, if B is a Fell bundle over G and we put H = { e } and K = G, this says B is amenable if G is amenable.Proof. The defining property for the Ψ maps and the notation adopted in 3.2 impliesthe diagram C ∗ K ( B ) Ψ B GK / / q B KH (cid:15) (cid:15) B ( C ∗ ( B ) ⊗ C ∗ K ( G )) ⊗ q GKH (cid:15) (cid:15) C ∗ H ( B ) Ψ B GH / / B ( C ∗ ( B ) ⊗ C ∗ H ( G ))commutes. The horizontal arrows are isometries and the vertical arrow on the right isan isomorphism, thus q B KH is a C*-isomorphism. (cid:3) Amenability and reductions to normal subgroups.
The main goal of thissection is to show that if B is a Fell bundle over G and both H and K are normalsubgroups of G with H ⊂ K, then q B K H : C ∗ ( B K ) → C ∗ H ( B K ) is a C*-isomorphism ifand only if q B KH : C ∗ K ( B ) → C ∗ H ( B ) is a C*-isomorphism. To do this we first show thefollowing. Theorem 3.12.
Let B be a Fell bundle over G and consider two subgroups of G such that H ⊂ K. If we define S := Λ H B | B K : B K → B ( L H ( B )) and there exists a*-representation π : C ∗ H ( B K ) → B ( L H ( B )) such that (3.13) L ( B K ) e S / / χ B K (cid:15) (cid:15) B ( L H ( B )) C ∗ ( B K ) q B KH / / C ∗ H ( B K ) π O O commutes, then π is unique, faithful, non degenerate, π ( C ∗ H ( B K )) ⊂ B ( C ∗ H ( B )) and π isthe unique non degenerate *-representation such that π (Λ H B K b ) = Λ H B b for all b ∈ B K , with π : B ( C ∗ H ( B K )) → B ( C ∗ H ( B )) being the unique *-representation extending π. Sucha map π exists if both H and K are normal in G. Proof.
By Theorem 2.15 C ∗ ( B H ) = C ∗ ( B + H ) , C ∗ ( B K ) = C ∗ ( B + K ) and C ∗ (( B K ) + H ) = C ∗ ( B + H ) = C ∗ ( B H ) . Assume π exists. Then (3.13) determines π in the dense set q B K H ( χ B K ( L ( B K )) and so it is unique. The diagram also implies the image of π isthe closure of e S ( L ( B K )) , which is a non degenerate *-subalgebra of B ( L H ( B )) because S is non degenerate. Thus π is non degenerate. The inclusion π ( C ∗ H ( B K )) ⊂ B ( C ∗ H ( B ))follows by regarding S as a *-representation of B K in B ( C ∗ H ( B )) . To prove the identity π (Λ H B K b ) = Λ H B b (for all b ∈ B ) we name P the *-representation B K → B ( L H ( B )) given by b π (Λ H B K b ) . Then(3.14) e P = π ◦ (Λ H B K | B K ) e = π ◦ q B K H ◦ χ B K = π ◦ q B K H ◦ χ B K = e S. Thus P and S have the same integrated form and P = S, this being the identity wewanted to prove. In fact (3.14) can also be used to show that any non degenerate NDUCTION, ABSORPTION AND AMENABILITY 35 *-representation π : C ∗ H ( B K ) → B ( L H ( B )) satisfying π (Λ H B K b ) = Λ H B b (for all b ∈ B )will also satisfy π ◦ q B K H ◦ χ B K = e S, making (3.13) a commutative diagram.To show that π is faithful recall it is contractive and that the induction in stages forFell bundles 2.62 gives S = Λ H B | B K = Ind B K (Λ H B K ) | B K . Then by Lemma 3.4 we have,for all f ∈ L ( B K ) , (3.15) k e Λ H B K f k = k q B K H ◦ χ B K ( f ) k ≥ k π ( q B K H ◦ χ B K ( f )) k = k e S f k ≥ k e Λ H B K f k . Thus π is isometric on a dense set, and it must be faithful.The discussion above implies that π exists if and only if k e S f k ≤ k e Λ H B K f k for all f ∈ C c ( B K ) . This is what we will now prove assuming both H and K are normal in G. Since K is normal in G we have ∆ G ( s ) = ∆ K ( s ) for all s ∈ K, in particular thisholds for all s ∈ H. Besides, H is normal in G and K, so ∆ G ( s ) = ∆ K ( s ) = ∆ H ( s )for all s ∈ H. Let Γ GK : G → (0 , + ∞ ) be the function such that for all a ∈ C c ( K )and r ∈ G, R K a ( trt − ) d K r = Γ GK ( t ) R K a ( r ) d K r (see [7, III 8.3]). We define Γ GH analogously, viewing H as a normal subgroup of G. We consider the left invariant Haarmeasures of
G, K and
G/K have been chosen so that Z G a ( r ) d G r = Z G/K Z K a ( rt ) d K t d G/K ( rK ) , ∀ a ∈ C c ( G ) . Fix f ∈ C c ( B K ) and take g ∈ C c ( B ) . Recall that L H ( B ) is constructed out of C c ( B )by using the C c ( B H ) − valued pre inner product [ g, h ] = p GH ( g ∗ ∗ h ) (with g, h ∈ C c ( B ) , see Remark 2.22). We want to prove that(3.16) h g, e S f ∗ ∗ f ( g ) i ≤ k e Λ H B K f ∗ ∗ f kh g, g i . Take a non degenerate *-representation T : B → B ( Y ) such that the integrated form e T : L ( B ) → B ( Y ) factors via a faithful *-representation χ B T : C ∗ ( B ) → B ( Y ) . Recallthat L G ( B ) ≡ C ∗ ( B ) and this identification gives a unitary equivalence L G ( B ) ⊗ T Y ≈ Y. Now Corollary 3.8 says the restriction T | B H : B H → B ( Y ) integrates to a*-representation L ( B H ) → B ( Y ) that factors through a faithful *-representation of ρ : C ∗ ( B H ) → B ( Y ) . Thus to prove (3.16) it suffices to show that(3.17) h ξ, ρ ( h g, e S f ∗ ∗ f ( g ) i ) ξ i ≤ k e Λ H B K f ∗ ∗ f kh ξ, ρ ( h g, g i ) ξ i , for all ξ ∈ Y. By the definition of ρ the identity above is equivalent to Z H Z K h ξ, T p GH ( g ∗ ∗ ( f ∗ ∗ f ( t ) g ))( s ) ξ i d K td H s ≤ k e Λ H B K f ∗ ∗ f k Z H h ξ, T p GH ( g ∗ ∗ g )( s ) ξ i d H s. By construction the left hand side of (3.12) is(3.18) Z H Z K Z G h T g ( r ) ξ, T f ∗ ∗ f ( t ) T g ( t − rs ) ξ i d G r d K t d H s == Z G/K Z K Z H Z K h T g ( rp ) ξ, T f ∗ ∗ f ( t ) T g ( t − rps ) ξ i d K t d H s d K p d G/K ( rK )= Z G/K Z K Z H Z K Γ GK ( r ) h T g ( pr ) ξ, T f ∗ ∗ f ( t ) T g ( t − prs ) ξ i d K t d H s d K p d G/K ( rK )= Z G/K Z K Z H Z K Γ GK ( r )Γ GH ( r ) h T g ( pr ) ξ, T f ∗ ∗ f ( t ) T g ( t − psr ) ξ i d K t d H s d K p d G/K ( rK ) . What we want to do now is to describe the inner triple integral in the last term aboveas an inner product. To do this we fix a coset rK, we even consider r fixed.Let κ : H → B be the trivial representation, set W := L H ( K ) ⊗ κ C , U := Ind KH ( κ )and consider the map L : C c ( K, Y ⊗ K C ) → B ( Y ⊗ K W ) constructed in the proof ofTheorem 2.44. Recall that Y ⊗ K C = ℓ ( K ) ⊗ Y ⊗ C . Given any u, v ∈ C c ( K ) , η, ζ ∈ Y and q, z ∈ K we consider the elements ϕ := u ⊙ ( δ z ⊗ η ⊗
1) and θ := v ⊙ ( δ z ⊗ ζ ⊗ C c ( K, Y ⊗ K C ) . Then h L ( ϕ ) , (1 ℓ ( K ) ⊗ ( T ⊗ U )) f ∗ ∗ f ( t ) L ( θ ) i == h δ q ⊗ η ⊗ ( u ⊗ κ , (1 ℓ ( K ) ⊗ ( T ⊗ U )) f ∗ ∗ f ( t ) ( δ z ⊗ ζ ⊗ ( v ⊗ κ i = h δ q ⊗ η ⊗ ( u ⊗ κ , δ z ⊗ T f ∗ ∗ f ( t ) ζ ⊗ ( U HKt ( v ) ⊗ κ i = h δ q , δ z ih η, T f ∗ ∗ f ( t ) ζ ih u ⊗ κ , U HKt ( v ) ⊗ κ i = h δ q , δ z ih η, T f ∗ ∗ f ( t ) ζ i Z K Z H u ( p ) v ( t − p − s ) d K p d H s = Z K Z H h ϕ ( p ) , (1 ⊗ ℓ ( K ) ⊗ T f ∗ ∗ f ( t ) ⊗ θ ( t − p − s ) i d K p d H s. The first and last terms of the identities above are additive and continuous in theinductive limit topology with respect to the variables ϕ and θ. Thus by continuity weget that h L ( ϕ ) , (1 ℓ ( K ) ⊗ ( T ⊗ U )) f ∗ ∗ f ( t ) L ( θ ) i == Z K Z H h ϕ ( p ) , (1 ⊗ ℓ ( K ) ⊗ T f ∗ ∗ f ( t ) ⊗ θ ( t − p − s ) i d K p d H s for all ϕ, θ ∈ C c ( K, Y ⊗ K C ) . Define h r ∈ C c ( K, Y ⊗ K C ) by h r ( s ) = δ r ⊗ T g ( sr ) ξ ⊗ Γ / GK ( r )Γ / GH ( r ) . Then the identityabove and Corollary 2.56 imply Z K Z H Z K Γ GK ( r )Γ GH ( r ) h T g ( pr ) ξ, T f ∗ ∗ f ( t ) T g ( t − psr ) ξ i d K t d H s d K p == Z K Z H Z K h h r ( p ) , (1 ⊗ ℓ ( K ) ⊗ T f ∗ ∗ f ( t ) ⊗ h r ( t − ps ) i d K p d H s d K t = Z K h L ( h r ) , (1 ℓ ( K ) ⊗ ( T ⊗ U )) f ∗ ∗ f ( t ) L ( h r ) i d K t = h L ( h r ) , (1 ℓ ( K ) ⊗ ( T ⊗ U )) e f ∗ ∗ f L ( h r ) i≤ k ( T ⊗ U ) e f ∗ ∗ f kh L ( h r ) , L ( h r ) i ≤ k e Λ HKf ∗ ∗ f kh L ( h r ) , L ( h r ) i≤ k e Λ HKf ∗ ∗ f k Z K Z H Γ GK ( r )Γ GH ( r ) h T g ( pr ) ξ, T g ( psr ) ξ i d H s d K p NDUCTION, ABSORPTION AND AMENABILITY 37
If we now go back to (3.18) and (3.17) and use the inequality we obtained before weget h ξ, ρ ( h g, e S f ∗ ∗ f ( g ) i ) ξ i = Z H Z K h ξ, T p GH ( g ∗ ∗ ( f ∗ ∗ f ( t ) g ))( s ) ξ i d K td H s = Z G/K Z K Z H Z K Γ GK ( r )Γ GH ( r ) h T g ( pr ) ξ, T f ∗ ∗ f ( t ) T g ( t − psr ) ξ i d K t d H s d K p d G/K ( rK ) ≤ k e Λ HKf ∗ ∗ f k Z G/K Z K Z H Γ GK ( r )Γ GH ( r ) h T g ( pr ) ξ, T g ( psr ) ξ i d H s d K p d G/K ( rK ) ≤ k e Λ HKf ∗ ∗ f k Z G/K Z K Z H Γ GK ( r )Γ GH ( r ) h T g ( pr ) ξ, T g ( prr − sr ) ξ i d H s d K p d G/K ( rK ) ≤ k e Λ HKf ∗ ∗ f k Z G/K Z K Z H Γ GK ( r ) h T g ( rr − pr ) ξ, T g ( rr − prs ) ξ i d H s d K p d G/K ( rK ) ≤ k e Λ HKf ∗ ∗ f k Z G/K Z K Z H h T g ( rp ) ξ, T g ( rps ) ξ i d H s d K p d G/K ( rK ) ≤ k e Λ HKf ∗ ∗ f k Z G Z H h T g ( r ) ξ, T g ( rs ) ξ i d H sd G r ≤ k e Λ HKf ∗ ∗ f k Z H h ξ, T g ∗ ∗ g ( s ) ξ i d H s ≤ k e Λ HKf ∗ ∗ f kh ξ, ρ ( h g, g i ) ξ i . Thus (3.16) holds and the proof is complete. (cid:3)
We now can prove one of the main results of this article.
Theorem 3.19.
Let B be a Fell bundle over G and take normal subgroups H ⊂ K ⊂ N ⊂ G. Define the maps q B KH : C ∗ K ( B ) → C ∗ H ( B ) and q B N KH : C ∗ K ( B N ) → C ∗ H ( B N ) as inProposition 3.1. If q B KH is a C*-isomorphism then q B N KH is so (and the converse holds,by Proposition3.1, if K = N ). In particular,(1) If H = { e } , then the fact of the canonical map q B KH : C ∗ K ( B ) → C ∗ r ( B ) being aC*-isomorphism implies q B N KH : C ∗ K ( B N ) → C ∗ r ( B N ) is so.(2) If B is amenable then so it is B N . Proof.
By Theorem 3.12 there exists non degenerate and faithful *-representations π K : C ∗ K ( B N ) → B ( C ∗ K ( B )) and π H : C ∗ H ( B N ) → B ( C ∗ H ( B )) such that π (Λ K B N b ) = Λ K B b and π H (Λ H B N b ) = Λ H B b for all b ∈ B N . We claim the diagram below commutes: C ∗ K ( B N ) π K / / q B NKH (cid:15) (cid:15) B ( C ∗ K ( B )) q B KH (cid:15) (cid:15) C ∗ H ( B N ) π H / / B ( C ∗ H ( B ))Indeed, for all f ∈ L ( B N ) we have q B KH ◦ π K ( χ B N ( f )) = q B HK ( e Λ K B f ) = ( q B HK ◦ Λ K B ) e f = e Λ H B f = ( π H ◦ Λ H B N ) e f = π H ◦ q B N KH ( χ B K ( f )) . Thus q B HK ◦ π K = π H ◦ q B N KH because both maps are continuous and agree on a denseset.Now, if q B KH is a C*-isomorphism then π H ◦ q B N KH is faithful. This implies q B N KH isfaithful and also a C*-isomorphism (recall it is surjective). (cid:3) C*-completions of Banach *-algebraic bundles
In this section we deal with C*-completions of a Banach *-algebraic bundle (otherthat the bundle C*-completion). We will use Definitions 1.4 and 1.5, the notation weadopted after them and the definition of the integrated form of a morphism of Banach*-algebraic bundles.
Proposition 4.1.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle and consider twoC*-completions, ι : B → A and κ : B → C . Then there exists a morphism ρ : ι → κ ifand only if k κ ( b ) k ≤ k ι ( b ) k for all b ∈ B e . In fact the morphism is unique (if it exists)and it is an isomorphism if and only if k ι ( b ) k = k κ ( b ) k for all b ∈ B e . Proof. If ρ : ι → κ is a morphism then k ρ k ≤ k κ ( b ) k = k ρ ( ι ( b )) k ≤k ι ( b ) k for all b ∈ B e . Assume, conversely, that k κ ( b ) k ≤ k ι ( b ) k for all b ∈ B e . Then for all b ∈ B we have k κ ( b ) k = k κ ( b ∗ b ) k / ≤ k ι ( b ∗ b ) k / = k ι ( b ) k . This last inequality, together with the factthat ι ( B t ) is a dense subspace of A t , implies the existence (for all t ∈ G ) of a uniquecontinuous linear map ρ t : A t → C t such that ρ t ( ι ( b )) = κ ( b ) for all b ∈ B t . Let ρ = { ρ t } t ∈ G : C → D be the unique extension of all the maps ρ t . Then Remark 1.6together with [7, II 13.16] implies ρ is continuous. It is also multiplicative and preservesthe involution when restricted to the dense set ι ( B ) , thus ρ : ι → κ is a morphism andit is unique because the condition ρ ◦ ι = κ determines ι in the dense set ι ( B ) . Theisomorphism claim follows immediately. (cid:3)
The Proposition above motivates the following Definition.
Definition 4.2.
Given a Banach *-algebraic bundle B = { B t } t ∈ G and a C*-completion ρ : B e → A, a ρ − completion of B is a C*-completion ι : B → A such that ι e = ρ. We now combine the induction process with the existence of particular C*-completionof Banach *-algebraic bundles.
Theorem 4.3.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle and let π : B e → A bea C*-completion, which we regard as a *-representation π : B e → A ⊂ B ( A ) . Then thefollowing are equivalent:(1) There exists a π − completion of B . (2) π is B− positive and π ( c ∗ b ∗ bc ) ≤ k π ( b ∗ b ) k π ( c ∗ c ) for all b, c ∈ B . (3) For every b, c ∈ B it follows that ≤ π ( c ∗ b ∗ bc ) ≤ k π ( b ∗ b ) k π ( c ∗ c ) . (4) π is B− positive and k Ind B{ e } ( π ) b k ≤ k π ( b ) k for all b ∈ B e . (5) π is B− positive and k Ind B{ e } ( π ) b k = k π ( b ) k for all b ∈ B e . (6) There exists a non degenerate *-representation T : B → B ( Y ) ( Y being a Hilbertspace) such that k π ( b ) k = k T b k for all b ∈ B e . (7) There exists a *-representation T : B → B ( Y A ) such that k π ( b ) k = k T b k for all b ∈ B e . Proof.
By Theorem 2.15 π is B− positive if and only if π ( b ∗ b ) ≥ b ∈ B . Thus(2) implies (3) and the existence of approximate units guarantees the converse.If (1) holds, then π is B− positive because for all b ∈ B we have π ( b ∗ b ) = ι ( b ) ∗ ι ( b ) ≥ . If we replace the topology of G by the discrete one, then A becomes a Fell bundle overa discrete group and using [5, Lemma 17.2] we deduce that a ∗ b ∗ ba ≤ k b ∗ b k a ∗ a (in A e )for all a, b ∈ A (no matter which topology we consider on G ). Thus for all b, c ∈ B wehave π ( c ∗ b ∗ bc ) = ι ( c ) ∗ ι ( b ) ∗ ι ( b ) ι ( c ) ≤ k ι ( b ∗ b ) k ι ( c ) ∗ ι ( c ) = k π ( b ∗ b ) k π ( c ∗ c ) . This shows (1)implies both (2) and (3).
NDUCTION, ABSORPTION AND AMENABILITY 39
We now assume (2) holds and will prove (4). Clearly π is B− positive. Noticing thatthe inner product of L e ( B ) ⊗ π A is given by h f ⊗ π a, g ⊗ π b i = R G a ∗ π ( f ( t ) ∗ g ( t )) b dt, using (2) it follows that for all ζ = P nj =1 f j ⊗ π π ( a j ) ∈ L e ( B ) ⊗ π A (with a j ∈ B e ) andall b ∈ Bh Ind B e ( π ) b ζ , Ind B e ( π ) b ζ i = Z G π (( n X j =1 a j f j ( t )) ∗ b ∗ b ( n X k =1 a k f k ( t ))) ≤ k π ( b ∗ b ) kh ζ , ζ i . Since π ( B e ) dense in A it follows that k Ind B e ( π ) b k ≤ k π ( b ∗ b ) k . If we now assume (4) holds and take a non degenerate and faithful *-representation ρ : A → B ( Z ) , then recalling that the abstract induced representation Ind B e ( π ⊗ ρ B e ↑B ( π ⊗ ρ
1) andusing [7, XI 11.3] we get that k π ( a ) k = k ( π ⊗ ρ a ) k ≤ k Ind B e ↑B ( π ⊗ ρ a k = k Ind B e ( π ⊗ ρ a k = k Ind B e ( π ) a k for all a ∈ B e . Then (5) follows and it implies (6) with T = Ind B e ( π ⊗ ρ k k T : B → R , b
7→ k T b k , instead of the norm k k c used by Fell.When Fell says “form the Hilbert direct sum T of enough *-representations of B so that k T b k = k b k c ” just consider the *-representation T given by claim (7). This produces aC*-completion ι : B → C in such a way that C e is naturally isomorphic to the closureof T ( B e ) . But claim (7) implies the existence of a unique isomorphism of C*-algebras φ : A → C e such that φ ◦ π = ι | B e . Then we may replace C e by A in C = { C t } t ∈ G and ι e by π in ι = { ι t } t ∈ G to get a C*-completion as in claim (1). (cid:3) Remark . The Theorem above implies that if B is a Fell bundle then there exists a*-representation T : B → B ( Y ) with k T b k = k b k for all b ∈ B e . Hence, for all b ∈ B , k T b k = k T b ∗ b k / = k b ∗ b k / = k b k . This is the property used by Fell in [7, VIII 16.10]to prove every Fell bundle is it’s own bundle C*-completion.
Remark . Let B = { B t } t ∈ G be aBanach *-algebraic bundle and let ι : B → C be it’s bundle C*-completion. For anyother C*-completion κ : B → A there exists a *-representation T : A → B ( Z ) suchthat k T a k = k a k for all a ∈ A . Consequently, T ◦ κ is a *-representation of B andthe construction of C implies k κ ( b ) k = k T ◦ κ ( b ) k ≤ k ι ( b ) k for all b ∈ B e . ThenProposition 4.1 implies the existence of a unique morphism ρ : C → A . So the bundleC*-completion is the universal C*-completion of B , and we denote it ι u : B → B u . Itthen follows immediately that every Fell bundle is it’s own bundle C*-completion (asalready noticed by Fell).Considering a fixed group G and a subgroup H of G, the proposition below can beused to construct functors( B ρ / / C ) ( C ∗ ( B + H ) χ ρ + H / / C ∗ ( C + H ) )( B ρ / / C ) ( C ∗ H ( B ) χ ρH / / C ∗ H ( C ) )from the category of Banach *-algebraic bundles over G to the category of C*-algebras. Proposition 4.6.
Let B = { B t } t ∈ G and C = { C t } t ∈ G be Banach *-algebraic bundles, H ⊂ G a subgroup and ρ : B → C a morphism of Banach *-algebraic bundles. Denote ρ H : B H → C H the morphism such that ρ H ( b ) = ρ ( b ) for all b ∈ B H . If either B is saturated or H is normal in G, then there exists unique morphism of *-algebras χ ρ + H and χ ρH such that the following diagrams commute L ( B H ) f ρ H / / χ B + H (cid:15) (cid:15) L ( C H ) χ C + H (cid:15) (cid:15) C ∗ ( B + H ) χ ρ + H / / C ( C + H ) C ∗ ( B ) χ ρ / / q B H (cid:15) (cid:15) C ∗ ( C ) q C H (cid:15) (cid:15) C ∗ H ( B ) χ ρH / / C ∗ H ( C )(4.7) In case χ ρ + H is faithful then so it is χ ρH . If ρ : B → C is a C*-completion then both χ ρ + H and χ ρH are surjective maps and they are C*-isomorphism if ρ is the canonical mapfrom a Banach *-algebraic bundle to it’s bundle C*-completion.If both B and C are Fell bundles and H = { e } , the statements above imply the uniqueextension of e ρ : L ( B ) → L ( C ) to a morphism of C*-algebras C ∗ r ( B ) → C ∗ r ( C ) is faithfulwhenever ρ e : B e → C e is so (because ρ e = χ ρ H ).Proof. The uniqueness of the morphism follows from the commutativity of the diagramsand the fact that the both χ B + H and q B H have dense ranges. To prove the existence of χ ρ + H we may regard C ∗ ( C + H ) as a non degenerate C*-subalgebra of B ( Y ) , for someHilbert space Y. This entails the existence of a non degenerate and C− positive *-representation T : C H → B ( Y ) such that the *-representation χ C + H T : C ∗ ( C + H ) → B ( Y ) isjust the inclusion map. Then T ◦ ρ H is B− positive by Theorem 2.15 and there exists amorphism of *-algebras χ B + H T : C ∗ ( B + H ) → B ( Y ) such that χ B + H T ◦ χ B + H = ( T ◦ ρ H ) e = e T ◦ f ρ H . Thus χ B + H T ( C ∗ ( B + H )) ⊂ C ∗ ( C + H ) and it suffices to set χ ρ + H := χ B + H T to prove the existenceof the morphism χ ρ + H with the desired properties.By Lemma 3.4 there exists a non degenerate *-representation T : C → B ( Y ) such thatthe restriction S := T | C H is non degenerate and χ C + H S : C ∗ ( C + H ) → B ( Y ) is faithful. Recallthat, by construction, χ C + H S ◦ χ C + H = e S. Let κ : H → C be the trivial representation, set U := Ind GH ( κ ) and consider the *-representation π T κ : C ∗ H ( C ) → B ( Y ⊗ ( L H ( G ) ⊗ κ C ))of Corollary 2.56. Then, by Theorem 2.44, k e Λ H C f k = k e Λ H C f ⊗ S ⊗ κ k = k Ind C H ( S ⊗ κ ) f k ≤ k ( T ⊗ U ) e f k = k π T κ ( e Λ H C f ) k ≤ k e Λ H C f k for all f ∈ L ( C ) . We conclude that π T κ is faithful.Although the composition T ◦ ρ : B → B ( Y ) may be degenerate, by restricting toit’s essential space and applying Corollary 2.56, we can construct a *-representation π ( T ◦ ρ ) κ : C ∗ H ( B ) → B ( Y ⊗ ( L H ( G ) ⊗ κ C )) such that π ( T ◦ ρ ) κ ◦ e Λ H B = ( T ◦ ρ ⊗ U ) e . Since π T κ ◦ e Λ H C ◦ e ρ = ( T ◦ ρ ⊗ U ) e = π ( T ◦ ρ ) κ ◦ e Λ H B , for all f ∈ L ( B ) we have(4.8) k e Λ H C e ρ ( f ) k = k π T κ ◦ e Λ H C ◦ e ρ ( f ) k = k π ( T ◦ ρ ) κ ( e Λ H B f ) k ≤ k e Λ H B f k . Then we can construct a morphism of *-algebras e Λ H B ( L ( B )) → C ∗ H ( C ) mapping e Λ H B f to e Λ H C e ρ ( f ) and this morphism if contractive with respect to the C*-norm inheritedfrom C ∗ H ( B ) . Thus the morphism has a unique extension to a morphism of C*-algebras χ ρH : C ∗ H ( B ) → C ∗ H ( C ) such that χ ρH ◦ e Λ H B = e Λ H C ◦ e ρ. Hence, for all f ∈ L ( B ) , χ ρH ◦ q B H ( e Λ G B f ) = χ ρH ◦ e Λ H B f = e Λ H C ◦ e ρ ( f ) = q C H ( e Λ G C ◦ e ρ ( f )) = q C H ◦ χ ρ ( e Λ G B f ) , and it follows that χ ρH ◦ q B H = q C H ◦ χ ρ . NDUCTION, ABSORPTION AND AMENABILITY 41
Assume χ ρ + H is faithful. Then χ C + H S ◦ χ ρ + H ◦ χ B + H = χ C + H S ◦ χ C + H ◦ f ρ H = e S ◦ f ρ H = ( T ◦ ρ | B H ) e , and this implies k χ B + H ( f ) k = k ( T ◦ ρ | B H ) e f k = k [( T ◦ ρ | B H ) ⊗ κ ] e f k for all f ∈ L ( B H ) . This implies the inequality in (4.8) is an equality and, in this case, our constructionof χ ρH implies it is faithful (to prove this claim follow our proof of the fact that π T κ isfaithful).Suppose ρ : B → C is a C*-completion. Then both e ρ and f ρ H have dense images andthis implies χ ρ + H and χ ρH have dense images and hence are surjective maps. (cid:3) Cross sectional bundles, C*-completions and induction.
Let’s fix, for therest to this section, a Banach *-algebraic bundle B = { B t } t ∈ G (with a strong approxi-mate unit, of course) and also a closed normal subgroup N of G. We will briefly recallthe main properties of the L − cross sectional bundle over G/N derived from B , thedetailed construction can found in [7, VIII 6].We regard the quotient group G/N = { tN : t ∈ G } as a LCH topological groupwith the quotient topology. Given a coset α = tN ( t ∈ G ) let ν α be the regular Borelmeasure on tN such that R α f ( x ) dν α ( x ) = R N f ( tx ) d N x for all f ∈ C c ( tN ) . Here d N x isthe integration with respect to a (fixed) left invariant Haar Measure of N. There is noambiguity in the definition of ν α because the left invariance of d N implies the function tN → R , r R N f ( rx ) d N x, is constant.For every function f ∈ C c ( G ) we define f : G → C as f ( t ) := R N f ( tx ) d N x = R tN f ( x ) dν tN ( x ) . It can be shown that f is continuous and constant in the cosets, soit defines a function f ∈ C ( G/N ) that vanishes outside the projection of supp( f ) on G/N.
Then, by construction, Z G/N f ( x ) d G/N x = Z G/N d G/N tN Z N f ( ts ) d N s. Throughout this work we assume the left invariant Haar measures of
G, N and
G/N are normalized in such a way that for all f ∈ C c ( G )(4.9) Z G f ( t ) d G t = Z G/N d G/N tN Z N f ( ts ) d N s, exactly as in [7, VIII 6.7]. In case G is a product H × K, N = H and K = ( H × K ) /H ;we meet this requirement by considering the product measure d G ( r, s ) = d H r × d K s. By [7, VIII 6.5] there exists a unique continuous homomorphism Γ : G → (0 , + ∞ )such that Z N f ( xyx − ) d N y = Γ( x ) Z N f ( y ) d N y, ∀ x ∈ G, f ∈ C c ( N ) . In case G is a product group H × K and N = H one has Γ( r, s ) = ∆ H ( r ) , where ∆ H is the modular function of H. The L − partial cross sectional bundle over G/N derived from B , C = { C α } α ∈ G/N , isdetermined by the following properties: • For every α ∈ G/N, if B α = { B t } t ∈ α is the reduction of B to α, then C α is thecompletion of C c ( B α ) with respect to the norm k f k = R α k f ( t ) k dν α ( t ) . • For every r, s ∈ G, f ∈ C c ( B rN ) and g ∈ C c ( B sN ) , the product f ∗ g ∈ C c ( B rsN ) ⊂ C rsN and the involution f ∗ ∈ C c ( B r − N ) are determined by f ∗ g ( x ) = R rN f ( y ) g ( y − x ) dν rN ( y ) and f ∗ ( z ) = Γ( z ) − f ( z − ) ∗ , for all x ∈ rsN and z ∈ r − N. • Given f ∈ C c ( B ) , if f | : G/N → C is given by the restriction f | ( α ) := f | α , then f | is a continuous cross section. Notation 4.10.
The L − partial cross sectional bundle over G/N derived from B willbe denoted L ( B , N ) = { L ( B α ) } α ∈ G/N . This is more than just a notation because ifgiven the coset α ∈ G/H we denote by B α the reduction of B to α , then B α is a Banachbundle and L ( B α ) is (constructively and symbolically) the completion of C c ( B α ) withrespect to k k . The usual L − cross sectional algebra of B may be regarded (notationallyand concretely) as L ( B , G ) and the bundle B itself as L ( B , { e } ) . Remark . Our conventions (C) state that B has a strong approximate unit, this isalso the case for L ( B , N ) by [7, VIII 6.9]. Remark . The set { f | : f ∈ C c ( B ) } ⊂ C c ( L ( B , N )) is dense in C c ( L ( B , N )) withrespect to the inductive limit topology by [7, II 14.6] and, consequently, it is dense in L ( L ( B , N )) . Remark . It is shown in [7, VIII 6.7] that there exists a unique isometric isomor-phism of Banach *-algebras Φ : L ( B ) → L ( L ( B , N )) such that Φ( f ) = f | , for all f ∈ C c ( B ) . Given a *-representation T : B → B ( Y A ) we can follow [7, VIII 15.9] and construct a(unique) *-representation(4.14) L ( T, N ) : L ( B , N ) → B ( Y A )such that for every coset α ∈ G/N, f ∈ C c ( B α ) and ξ ∈ Y A ,L ( T, N ) f ξ = Z G T f ( t ) ξ dν α ( t ) . If Φ : L ( B ) → L ( L ( B , N )) is the isomorphism of Remark 4.13, then f L ( T, N ) ◦ Φ = e T ; f L ( T, N ) being the integrated form of L ( T, N ) . In case we are given a non degenerate *-representation S : L ( B , N ) → B ( Y A ) , thecomposition e S ◦ Φ : L ( B ) → B ( Y A ) is a non degenerate *-representation that canbe disintegrated (uniquely) to a *-representation T : B → B ( Y A ) . Thus we obtain e T = e S ◦ Φ , implying both e T and T are non degenerate. Moreover, f L ( T, N ) ◦ Φ = e T = e S ◦ Φ . Thus f L ( T, N ) = e S and this implies L ( T, N ) = S. Any subgroup of
G/N can be expressed as
H/N for a unique subgroup H of G containing N. Then we obtain the identity of Banach *-algebraic bundles L ( B H , N ) ≡ { L ( B α ) } α ∈ H/N ≡ L ( B , N ) H/N ;and we have an isometric isomorphism of Banach *-algebrasΦ H : L ( B H ) → L ( L ( B H , N )) ≡ L ( L ( B , N ) H/N );which for H = G is just the isomorphism L ( B ) ∼ = L ( L ( B , N )) we have consideredbefore.The equivalence of the representations theories of B and L ( B , N ) now reduces to acorrespondence T ! L ( T, N ) between non degenerate *-representations of B H and L ( B H , N ) ≡ L ( B , N ) H/N . Given a *-representation T : B H → B ( Y A ) and a faithful *-representation ρ : A → B ( Z ) we have L ( T, N ) ⊗ ρ L ( T ⊗ ρ , N ) . Hence by Theorem 2.15 and [7, XI 12.7]the following are equivalent:(1) L ( T, N ) is L ( B , N ) − positive.(2) L ( T ⊗ ρ , N ) is L ( B , N ) − positive.(3) T ⊗ ρ B− positive.(4) T is B− positive. NDUCTION, ABSORPTION AND AMENABILITY 43
The equivalence of the claims above should be kept in mind, as we will use it quitefrequently without explicit mention.We want to stress two points. Firstly, when determining positivity of *-representationswe may restrict to the essential space (and assume non degeneracy) because we are as-suming B has a strong approximate unit. Secondly, [7, XI 12.7] is stated for nondegenerate *-representation and so can be extended to our context by the previouscomment. In fact Fell begins the proof of [7, XI 12.7] adopting “Rieffel’s formulationof the inducing process” meaning he is using the abstract inducing process, i.e. theprocess we have been working with. Hence, all the computations of [7, XI 12.7] holdverbatim replacing Ind B H ↑B by Ind B H (concrete by abstract) and even considering repre-sentations on Hilbert modules. In particular equation (6) in [7, pp 1164] becomes (inour notation)(4.15) f L ( T, N ) p G/NH/N (Φ( f ) ∗ ∗ Φ( g )) = f L ( T, N ) p G/NH/N (Φ( f ∗ ∗ g )) = e T p GH ( f ∗ ∗ g ) ;and holds for every *-representation T : B H → B ( Y A ) and f, g ∈ C c ( B ) . Here the p functions are the generalized restrictions for the groups indicated in the super and subindexes.Given a non degenerate L ( B , N ) − positive *-representation S : L ( B , N ) H/N → B ( Y )let T : B H → B ( Y ) be the (non degenerate) *-representation such that S = L ( T, N ) . We then get the following commutative diagram L ( L ( B , N ) H/N ) e S ≡ e L ( T,N ) * * π ◦ Φ H − ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ L ( B H ) e T / / Φ H o o π (cid:15) (cid:15) B ( Y ) C ∗ ( B + H ) π T ♥♥♥♥♥♥♥♥♥♥♥♥♥ We can arrange the T above so that π T is faithful. In such a situation we know(by [7, XI 12.7] or (4.15)) that π ◦ Φ H − ( p G/NH/N ( f ∗ ∗ f )) ≥ f, g ∈ C c ( L ( B , N )) . Then π ◦ Φ H − : L ( L ( B , N ) H/N ) → C ∗ ( B + H )is a C*-completion satisfying conditions (1) and (2) of Theorem 2.16, meaning that C ∗ ( L ( B , N ) + H/N ) ∼ = C ∗ ( B + H ) . By restricting the isometric *-isomorphism Φ : L ( B ) → L ( L ( B , N )) to C c ( B ) weobtain a unitary operator U : L H ( B ) → L H/N ( L ( B , N ))mapping f + I ∈ C c ( B ) /I to Φ( f ) + I ′ , I ′ being the null space of C c ( L ( B , N )) withrespect to the C ∗ ( L ( B , N ) + H/N ) − valued pre inner product of C c ( L ( B , N )) . This claimholds, ultimately, by (4.15). It then follows that the conjugation by U gives the unitaryequivalence of C*-algebras(4.16) C ∗ H ( B ) ∼ = C ∗ H/N ( L ( B , N )) ∼ = C ∗ H/N ( C ∗ ( B , N )) , where the C*-isomorphism on the right is that given by Proposition 4.6.We can now re-interpret (and extend) [7, XI 12.7] as the fact that for every B− positive*-representation T : B H → B ( Y A ) there exits a unitary U ⊗ L H ( B ) ⊗ T Y A → L H/N ( L ( B , N )) ⊗ L ( T,N ) Y A , mapping f ⊗ T ξ to Φ( f ) ⊗ L ( T,N ) ξ, for every elementary tensor f ⊗ T ξ. This operatorestablishes the unitary equivalence of *-representations(4.17) L (Ind B H ( T ) , N ) ∼ = Ind L ( B ,N ) H/N ( L ( T, N )) . The first consequence of the identity above is the following.
Proposition 4.18.
Assume B is a Fell bundle over G, N ⊂ G a normal subgroup and χ B N : L ( B N ) → C ∗ ( B N ) the universal C*-completion. Then the bundle C*-completionof L ( B , N ) , C = { C α } α ∈ G/N , is (isomorphic to) the χ B N − completion of L ( B , N ) inthe sense of Definition 4.2.Proof. Let ρ = { ρ α } α ∈ G/N : L ( B , N ) → C be the canonical morphism. By Proposi-tion 4.1 it suffices to show that k ρ N ( f ) k = k χ B N ( f ) k for all f ∈ L ( B N ) . The inequal-ity k ρ N ( f ) k ≤ k χ B N ( f ) k holds for all f ∈ L ( B N ) because C N is a C*-algebra and ρ N : L ( B N ) → C N a morphism of *-algebras.By Theorem 2.15 we can induce χ B N to a *-representation S := Ind L ( B ,N ) e ( χ B N ) of L ( B , N ) . Then Lemma 3.4 and the construction of the bundle C*-completion imply,for all f ∈ L ( B N ) , that k χ B N ( f ) k = k S f k ≤ k ρ N ( f ) k . Hence k χ B N ( f ) k ≤ k ρ N ( f ) k ≤ k χ B N ( f ) k and the proof is complete. (cid:3) It now makes sense to adopt the following.
Notation 4.19.
In the situation of the Proposition above, the bundle C*-completion of L ( B , N ) will be denoted C ∗ ( B , N ) = { C ∗ ( B α ) } α ∈ G/N . This makes sense because withthis notation the unit fibre of C ∗ ( B , N ) is, both symbolically and concretely, C ∗ ( B N ) . Previously in this section we obtained a C*-isomorphism C ∗ H ( B ) ∼ = C ∗ H/N ( L ( B , N )) , provided that H ⊂ G is a subgroup containing N. We now relate this fact to theamenability of L ( B , N ) . Proposition 4.20.
Consider a *-Banach algebraic bundle B = { B t } t ∈ G and subgroups N ⊂ H ⊂ K ⊂ G with N normal in G. Let
Φ : L ( B ) → L ( B , N ) be the isomorphismof Remark 4.13, ψ H : C ∗ H ( B ) → C ∗ H/N ( L ( B , N )) and ψ K : C ∗ K ( B ) → C ∗ K/N ( L ( B , N )) the C*-isomorphism of (4.16) and consider the quotient maps q for B and L ( B , N ) ofProposition 3.1. Then the following diagram commutes (4.21) L ( B ) e Λ K B / / Φ (cid:15) (cid:15) C ∗ K ( B ) q B KH / / ψ K (cid:15) (cid:15) C ∗ H ( B ) ψ H (cid:15) (cid:15) L ( B , N ) e Λ K/NL B ,N ) / / C ∗ K/N ( L ( B , N )) q L B ,N )( K/N )( H/N ) / / C ∗ H/N ( L ( B , N )) . In particular, q B KH is a C*-isomorphism if and only if q L ( B ,N )( K/N )( H/N ) is so. Setting K = G and H = N the preceding claim becomes: q B N : C ∗ ( B ) → C ∗ N ( B ) is a C*-isomorphism ifand only if q L ( B ,N ) e : C ∗ ( L ( B , N )) → C ∗ r ( L ( B , N )) is a C*-isomorphism (i.e. L ( B , N ) is amenable).Proof. The outer and inner left rectangles of 4.21 commute by the construction of the ψ maps and Proposition 3.1. Besides, the compositions of the q and χ maps in theinner left diagram have dense ranges. This forces the inner right diagram to commute.The rest of the proof follows immediately. (cid:3) NDUCTION, ABSORPTION AND AMENABILITY 45
At this point we have a complete understanding of the relation between the C*-algebras C ∗ H ( B ) and C ∗ K ( L ( B , N )) provided that N ⊂ H and K ⊂ G/N are subgroups.What about the C*-algebras C ∗ H ( B ) for H ⊂ N ?The following result may be used in conjunction with Proposition 3.1, Corollary 3.11or Theorem 3.19 to describe C ∗ r ( L ( B , N )) as C ∗ H ( B ) for some subgroup H ⊂ N. Proposition 4.22.
Let B = { B t } t ∈ G be a Banach *-algebraic bundle and N ⊂ G anormal subgroup. Then for any subgroup H ⊂ N there exists a unique morphism of*-algebras π H : C ∗ r ( L ( B , N )) → C ∗ H ( B ) such that the diagram below commutes (4.23) L ( L ( B , N )) e L (Λ H B ,N ) / / e Λ eL B ,N ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ C ∗ H ( B ) C ∗ r ( L ( B , N )) π H ♣♣♣♣♣♣♣♣♣♣♣ Moreover, π H is surjective and it is faithful if and only if q B NH : C ∗ N ( B ) → C ∗ H ( B ) is so. In particular, if B is a Fell bundle, H = { e } and we identify C ∗ r ( L ( B , N )) with C ∗ r ( C ∗ ( B , N )) as in Proposition 4.6, then π e : C ∗ r ( C ∗ ( B , N )) → C ∗ r ( B ) is a C*-isomorphism if and only if B N is amenable.Proof. It is implicit in the claim of the proposition that the image of f L (Λ H B , N ) : L ( L ( B , N )) → B ( C ∗ H ( B ))is contained in C ∗ H ( B ) . This is so because the image of e Λ H B : L ( B ) → B ( C ∗ H ( B )) iscontained in C ∗ H ( B ) and the diagram L ( B ) Φ x x ♣♣♣♣♣♣♣♣♣♣ e Λ H B % % ❑❑❑❑❑❑❑❑❑❑ L ( L ( B , N )) e L (Λ H B ,N ) / / B ( C ∗ H ( B ))commutes, with Φ being the isomorphism of Remark 4.13.By (4.21) and Proposition 3.1 the following diagram commutes: L ( L ( B , N )) L (Λ H B ,N ) / / χ L B ,N ) (cid:15) (cid:15) L (Λ N B ,N ) ' ' ◆◆◆◆◆◆◆◆◆◆◆ C ∗ H ( B ) C ∗ N ( B ) q B NH ♣♣♣♣♣♣♣♣♣♣♣ ψ N ' ' ◆◆◆◆◆◆◆◆◆◆◆ C ∗ ( L ( B , N )) q L B ,N ) e / / C ∗ r ( L ( B , N )) q B NH ◦ ( ψ N ) − O O and q L ( B ,N ) e ◦ χ L ( B ,N ) = e Λ eL ( B ,N ) . Thus we may define π H := q B NH ◦ ( ψ N ) − tomake (4.23) a commutative diagram. It is the unique with such property because (4.23)determines π H in a dense set. Besides, π H is surjective and our construction implies itis faithful if and only if q B NH is so.If B is a Fell bundle and H = { e } then, by Theorem 3.19, π H is a C*-isomorphism ⇔ q B NH is a C*-isomorphism ⇔ B N is amenable. (cid:3) Theorem 4.24.
Let B = { B t } t ∈ G be a Fell bundle and consider normal subgroups of G, H ⊂ N ⊂ G. Then the C*-completion q B N H ◦ χ B N : L ( B N ) → C ∗ H ( B N ) satisfies theequivalent conditions of Theorem 4.3 when considered as *-representation of the unitfibre of L ( B , N ) . If κ : L ( B , N ) → C ∗ H ( B , N ) is the q B N H ◦ χ B N − completion of L ( B , N ) , then there exists a unitary U : L H ( B ) → L e ( C ∗ H ( B , N )) with the following properties:(1) U ( f ) = e κ ◦ Φ( f ) , for all f ∈ C c ( B ) , with Φ : L ( B ) → L ( L ( B , N )) being thatRemark 4.13.(2) C ∗ H ( B ) = { U ∗ M U : M ∈ C ∗ r ( C ∗ H ( B , N )) } . (3) If ϕ : C ∗ H ( B ) → C ∗ r ( C ∗ H ( B , N )) is given by ϕ ( M ) = U M U ∗ , then the diagram C ∗ r ( L ( B , N )) π H w w ♣♣♣♣♣♣♣♣♣♣♣ χ κe ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ C ∗ H ( B ) ϕ / / C ∗ r ( C ∗ H ( B , N )) commutes; with χ κe : C ∗ r ( L ( B , N )) → C ∗ r ( C ∗ H ( B , N )) being the map provided byProposition 4.6 and π H the map of Proposition 4.22.Besides, the following are equivalent:(i) q B H : C ∗ ( B ) → C ∗ H ( B ) is a C*-isomorphism.(ii) q B N : C ∗ ( B ) → C ∗ N ( B ) and q B N H : C ∗ ( B N ) → C ∗ H ( B N ) are C*-isomorphisms.(iii) C ∗ H ( B , N ) is amenable and q B N H is a C*-isomorphism.(iv) C ∗ ( B , N ) is amenable and q B N H is a C*-isomorphism.(v) The morphism between the C*-completions ρ : C ∗ ( B , N ) → C ∗ H ( B , N ) , providedby Remark 4.5, is an isomorphism and C ∗ ( B , N ) is amenable.Proof. If in Theorem 3.12 we put K = N, then the *-representation S of that Theoremis (Λ H B | B N ) e = L (Λ H B , N ) | L ( B N ) . Thus (3.15) gives the identity k q B N H ◦ χ B N ( f ) k = k L (Λ H B , N ) f k for all f ∈ L ( B N ) . It isthen clear that q B N H ◦ χ B N satisfies condition (7) of Theorem 4.3. Let then κ : L ( B , N ) → C ∗ H ( B , N ) be the q B N H ◦ χ B N − completion of L ( B , N ) . We now will prove the existence of the unitary U : L H ( B ) → L e ( C ∗ H ( B , N )) such that U f = e κ ◦ Φ( f ) for all f ∈ C c ( B ) . Recall that we may view L H ( B ) as a completion of C c ( B ) (see Remark 2.22). To prove the existence of U take f, g ∈ C c ( B ) and note that: h e κ ◦ Φ( f ) , e κ ◦ Φ( g ) i L e ( C ∗ H ( B ,N )) = ( e κ ◦ Φ( f )) ∗ ∗ ( e κ ◦ Φ( g ))( N ) = e κ ◦ Φ( f ∗ ∗ g )( N )= ρ ( f ∗ ∗ g | N ) = f ∗ ∗ g | N = p GN ( f ∗ ∗ g ) = h f, g i L H ( B ) ;where the identity f ∗ ∗ g | N = p GN ( f ∗ ∗ g ) holds because ∆ G | N = ∆ N . It is then clearthat there exists a unique linear map U : L H ( B ) → L e ( C ∗ H ( B , N )) that preserves innerproducts and when restricted to C c ( B ) is given by f e κ ◦ Φ( f ) . Note that U issurjective because it is an isometry and it’s image contains the set e κ ◦ Φ( C c ( B )) , which isdense in the inductive limit topology in C c ( C ∗ H ( B , N )) and hence dense in L e ( C ∗ H ( B , N ))by Remark 2.21.For all f, g ∈ C c ( B ) we have e Λ eC ∗ H ( B ,N ) e κ ◦ Φ( f ) U g = ( e κ ◦ Φ( f )) ∗ ( e κ ◦ Φ( g )) = e κ ◦ Φ( f ∗ g ) = U e Λ H B f g. Thus e Λ eC ∗ H ( B ,N ) ( e κ ◦ Φ( f )) = U e Λ H B f U ∗ for all f ∈ L ( B ) and it follows that e Λ eC ∗ H ( B ,N ) ( L ( B , N )) = U e Λ H B ( L ( B )) U ∗ . NDUCTION, ABSORPTION AND AMENABILITY 47
Claim (2) of the present Theorem follows by taking closures on both sides of the identityabove.The construction of π H , ϕ and χ κe imply that for all f ∈ L ( B ) : π H ( e Λ eL ( B ,N )Φ( f ) ) = f L (Λ H B , N ) ◦ Φ( f ) = e Λ H B f = U ∗ e Λ eC ∗ H ( B ,N ) e κ ◦ Φ( f ) U = ϕ (cid:18) e Λ eC ∗ H ( B ,N ) e κ ◦ Φ( f ) (cid:19) = ϕ ( χ κe ( e Λ eL ( B ,N )Φ( f ) ));meaning that π H and ϕ ◦ χ κe agree when restricted to the image of e Λ eL ( B ,N ) . Thus, bycontinuity, π H = ϕ ◦ χ κe . Having shown claims (1) to (3) we now turn to prove theequivalence of the claims (i) to (v).By Proposition 3.1 claim (i) is equivalent to say both q B H : C ∗ ( B ) → C ∗ N ( B ) and q B KH : C ∗ K ( B ) → C ∗ H ( B ) are C*-isomorphisms; which in turn is equivalent to (ii) by Theo-rem 3.19. Claim (ii) together with Proposition 4.18 implies the C*-completions C ∗ H ( B N )and C ∗ ( B N ) of the unit fibre L ( B N ) of L ( B , N ) agree, thus the C*-completions C ∗ ( B , N ) and C ∗ H ( B , N ) agree by Proposition 4.1. By using the identifications C ∗ ( B ) ≡ C ∗ ( C ∗ ( B , N )) and C ∗ N ( B ) ≡ C ∗ r ( C ∗ ( B , N )) of Proposition 4.20, it follows that (ii)implies (iii). As explained before, the fact of q B N H being a C*-isomorphism implies C ∗ H ( B , N ) = C ∗ ( B , N ) . Thus (iii) and (iv) are equivalent and both imply (v). Fi-nally, by Propositions 4.1 and 4.18, claim (v) implies q B N H is a C*-isomorphism and, byProposition 4.20, that q B N is a C*-isomorphism. Hence (v) implies (ii) and the proof iscomplete. (cid:3) As a particular case of the Theorem above one may consider H = { e } and, as usual,write C ∗ r instead of C ∗{ e } . In doing so one obtains the equivalence of the following claims:(1) B is amenable.(2) C ∗ ( B , N ) and B N are amenable.(3) C ∗ r ( B , N ) and B N are amenable.(4) The morphism of C*-completions ρ : C ∗ ( B , N ) → C ∗ r ( B , N ) is an isomorphismand C ∗ ( B , N ) is amenable.At this point there is no much room left for applications and consequences of thetheory we have developed. Still, we want to give some example of how to use ourtheory. We start with to Corollaries that are known to hold for the subgroups H = { e } and H = G [1, 4]. These results are an extension to Fell bundles of the compatibilitybetween the induction from subgroups and any Morita equivalence of crossed productscoming from a Morita equivalence of actions (see the motivating examples of [1]). Corollary 4.25.
Let B = { B t } t ∈ G be a Fell bundle and A ⊂ B a Fell subbundle. If N ⊂ G is a normal subgroup and A N is hereditary in B N in the sense of [4], then C ∗ N ( A ) is C*-isomorphic to the closure of q B N ( L ( A )) in C ∗ N ( B ) . Proof.
By [4] we know C ∗ ( A N ) is (C*-isomorphic to) the closure of L ( A N ) in L ( B N ) . If we regard L ( A , N ) as a Banach *-algebraic subbundle of L ( B , N ) and then useProposition 4.18 to construct the C*-completions C ∗ ( A , N ) and C ∗ ( B , N ) , we then canview C ∗ ( A , N ) as a Fell subbundle of C ∗ ( B , N ) . It follows from [1, Proposition 3.2] thatthe inclusion L ( C ∗ ( A , N )) ֒ → L ( C ∗ ( B , N )) extends to an inclusion C ∗ r ( C ∗ ( A , N )) ֒ → C ∗ r ( C ∗ ( B , N )) . If we view L ( A ) = L ( L ( A , N )) as a *-subalgebra of L ( C ∗ ( A , N ))and identify C ∗ N ( A ) with C ∗ r ( C ∗ ( A , N )) (and do the same thing for B ) we see theinclusion L ( A ) ֒ → L ( B ) extends to an inclusion C ∗ N ( A ) ֒ → C ∗ N ( B ) , the image ofwhich is q B N ( L ( A )) . (cid:3) Corollary 4.26.
Let A and B be Fell bundles over G which are (Morita) equivalent inthe sense of [4]. Then for every normal subgroup N ⊂ G the C*-algebras C ∗ N ( A ) and C ∗ N ( B ) are Morita equivalent.Proof. Let X be an A − B− equivalence bundle and form the linking bundle L ( X ) as ex-plained in [4]. Then both A and B are hereditary Fell subbundles of L ( X ) , so both A N and B N are hereditary in C ∗ N ( L ( X )) and we may identify C ∗ N ( A ) ≡ q L ( X ) N ( C ∗ ( A )) and C ∗ N ( B ) ≡ q L ( X ) N ( C ∗ ( B )) . Define, as in [4], C ∗ ( X ) as the closure of L ( X ) ⊂ L ( L ( X )) in C ∗ ( L ( X )) . Then C ∗ ( X ) is a C ∗ ( A ) − C ∗ ( B ) − equivalence bimodule with bimodule struc-ture inherited from the canonical C ∗ ( L ( X )) − C ∗ ( L ( X )) − equivalence module structureof C ∗ ( L ( X )) . Since q L ( X ) : C ∗ ( L ( X )) → C ∗ N ( L ( X )) is a surjective and a morphism ofC*-algebras, it follows that C ∗ N ( X ) := q L ( X ) N ( C ∗ ( X )) is a C ∗ N ( A ) − C ∗ N ( B ) − equivalencebimodule. (cid:3) We close this article with a Corollary relating the weak approximation property(WAP) of [2] and our constructions. This last result is intended to give an idea of howto combine our characterization C ∗ N ( B ) = C ∗ r ( C ∗ ( B , N )) with the WAP. The readermay use the ideas we expose to produce other results of this sort. Corollary 4.27.
Let B be a Fell bundle over a discrete group G and N ⊂ G a normalsubgroup. Then the following are equivalent.(1) C ∗ ( B ) is nuclear.(2) C ∗ N ( B ) is nuclear.(3) C ∗ r ( B ) is nuclear.(4) B has the WAP and B e is nuclear.(5) C ∗ ( B N ) is nuclear and C ∗ ( B , N ) has the WAP.(6) C ∗ r ( B N ) is nuclear and C ∗ r ( B , N ) has the WAP.(7) Both B N and C ∗ ( B , N ) have the WAP and B e is nuclear.(8) Both B N and C ∗ r ( B , N ) have the WAP and B e is nuclear.If the conditions above hold, then both B and B N are amenable.Proof. Assume (1) holds, then (2) does so because C ∗ N ( B ) is a quotient of a nuclearC*-algebra. Recalling that C ∗ r ( B ) = C ∗{ e } ( B ) is a quotient of C ∗ N ( B ) , we conclude (2)implies (3). If (3) holds, then the existence of a conditional expectation from C ∗ r ( B )to B e [5] implies B e is nuclear and we then can use [2, Proposition 7.3] to conclude B has the WAP. The same Proposition can be used to prove (4) implies (1), then thethe first four claims are equivalent. This equivalence together with the fact that C ∗ ( B )is C*-isomorphic to C ∗ ( C ∗ ( B , N )) (and C ∗ r ( B ) to C ∗ r ( C ∗ r ( B , N ))) implies the first sixclaims are equivalent. Finally, this last equivalence can be used to prove all the eightclaims are equivalent. The proof ends after one recalls from [2] that the WAP impliesamenability. (cid:3) References [1] Fernando Abadie. Enveloping actions and Takai duality for partial actions.
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