Amenability, Nuclearity and Tensor Products of C ∗ -Algebraic Fell Bundles under the Unified Viewpoint of the Fell-Doran Induced Representation Theory
aa r X i v : . [ m a t h . OA ] A p r (First Draft) Amenability, Nuclearity and TensorProducts of C ∗ -Algebraic Fell Bundles under theUnified Viewpoint of the Fell-Doran InducedRepresentation Theory Weijiao HeApril 9, 2020
Abstract
In this paper we study amenability, nuclearity and tensor products of C ∗ -Fell bundles by the method of induced representation theory. Introduction C ∗ -algebraic Fell bundles over locally compact groups, which we denoted by B ,were defined and studied by Fell and Doran in [12]. A lot of definitions and the-orems on locally compact groups can be generalized on B . In [10], Exel and Ngstudied the amenability and the approximation property (AP) of B , which were thegeneralization of the amenability and approximation property of locally compactgroups. In the recent years, the problems about B which are in the connection withamenability, tensor products and nuclearity of C ∗ -algebras are studied under variedconditions: see Abadie, Buss and Ferraro [1] for the case that B is over locallycompact groups; Buss, Echterhoff and Willett [6] for the case that B is semidirectproduct bundles over locally compact groups; Abadie-Vicens [3], Abadie, Buss andFerraro [2], Ara, Exel, and Katsura [5], Exel [9], He [13], McKee, Skalski, Todorovand Turowska[17] for the case that B is over discrete groups; Lalonde [16] and Simsand Williams [19] [18] for the case that B is separable and is over secound countable groupoids ; Takeishi [20] for the case that B is over ´ Etale groupoids .The main objective of this paper is, letting B to be a C ∗ -algebraic Fell bundleover locally compact groups, to study the relations between the amenability of B ,tensor products of B and C ∗ -algebras, and the nuclearity of the full or regular C ∗ -algebras of B , by the method of induced representation theory developed by Felland Doran in [12]. Corollary 2.4 and Theorem 4.7 give partial solution to ClaireAnantharaman-Delaroche [4, Problem 9.2(d)], which is partially the motivation ofthis paper.In section 1, we give a brief review of the theory of induced representation of C ∗ -algebraic Fell bundles and Fell-Doran’s Imprimitivity Theorem.1n section 2, we prove that the Moraita equivalence constructed by Fell-Doran’sImprimitivity Theorem preserves the regular representations.In section 3, we develop the theory of the tensor products of C ∗ -algebraic Fellbundles and C ∗ -algebras. In the case that G is discrete, this theory is developed insome papers we referred in the first paragraph. Our treatment based on the inducedrepresentation theory is new, strikingly easier, and generalize the results which holdfor discrete bundles.In section 4, we give the definition of ultra-approximation property (UAP) of C ∗ -algebraic Fell bundles, which is weaker than the approximation property and is asufficient condition by which the bundle is amenable. Based on the results of Propo-sition 4.4 and Proposition 4.5, in Remark 4.6 we show that there are plenty “non-trivial” examples of C ∗ -algebraic bundles having UAP, particularly are amenable.Finally, we combine Proposition 4.4 and Proposition 4.5 with the results of section2 and section 3 to prove our main result, i.e. Theorem 4.7. Throughout this paper, G is a locally compact group and we choose once for all aHaar measure on G , R G is the left-regular representation of G , and H ⊂ G a closedsubgroup of G . In addition, we will choose once for all a continuous everywherepositive H -rho function ρ on G and denote by ρ ♯ the regular Borel measure on G/H constructed from ρ . Assume B is a C ∗ -algebraic Fell bundle over G with fibers { B x } x ∈ G , and let B H be the restricted bundle of B to H . Let L ( B ), L ( B ) and L ( B ) to denote the space of continuous cross-sections vanishing at infinite point,absolute integral cross-sections and squared-integrable cross-sections respectively.For details on the definition of C ∗ -algebraic Fell bundles over locally compact group G and the cross-sectional algebras we refer the reader to [12, § Chapter VIII].For ∗ -algebra A ( resp. ∗ -algebraic bundle B ), let T and S be ∗ -representationsof A ( resp. B ), we use symbol T ≤ S to indicate that S weakly contains T .If T is ∗ -representation of C ∗ -algebraic bundle B , we use symbol e T to denotethe integrated form of T (see [12, § VIII.12]). But sometimes we directly regard T as ∗ -representation of L ( G, B ) or C ∗ ( B ), i.e its integrated form.In this section we review some basic notions of the theory on induced represen-tations of C ∗ -algebraic Fell bundles which are treated in details in [12, § XI.9].Let S be a B - positive (see [12, § XI.8]) non-degenerate ∗ -representation of B H .Let Z α be the algebraic direct sum P ⊕ x ∈ α ( B x ⊗ X ( S )) of the algebraic tensor products B x ⊗ X ( S ), we introduce into Z α the conjugate-bilinear form ( , ) α by( b ⊗ ξ, c ⊗ η ) α = ( ρ ( x ) ρ ( y )) − / ( S c ∗ b ξ, η )( x, y ∈ α ; b ∈ B x ; c ∈ B y ; ξ, η ∈ X ( S )). One can form a Hilbert space Y α by factoring out from Z α the null space of ( , ) α and completing, and a Hilbertbundle Y over G/H with fibers Y α . Let κ α : Z α → Y α be the quotient map for each α ∈ G/H . For each c ∈ B x , there is continuous map τ c : Y → Y defined by τ c ( κ α ( b ⊗ ξ )) = κ xα ( cb ⊗ ξ ) ( b ∈ B y , y ∈ α ; ξ ∈ X ( S )) . b ∈ B ( α f ( α )) ( f : α τ b f ( x − α )) ( f ∈ L ( G/H ; Y ))is a bounded operator on the Hilbert space L ( G/H ; Y ), which we denote by T b , andthat b T b is non-degenerate ∗ -representation of B . We denote this representation T by Ind B H ↑ B ( S ), and say that T is induced from S .By [12, XI.11.7] every ∗ -representation S of B e is B -positive; if in addition B issaturated, by [12, XI.11.10] every ∗ -representation S of B H is B -positive. Therefore,in either of the case H = { e } or that B is saturated, Ind B H ↑ B ( S ) exists.Let ( G, M ) be a G -transformation group. By [12, § VIII.7] one can construct the
G, M transf ormation bundle D over G derived f rom B , whose fiber D x for each x ∈ G is C ( M ; B x ), i.e the set of continuous functions from M into B x vanishingat infinity point, with multiplication and involution given by( φψ )( m ) = φ ( m ) ψ ( x − m ) ,φ ∗ ( m ) = ( φ ( xm )) ∗ ( x, y ∈ G ; φ ∈ D x ; ψ ∈ D y ; m ∈ M ). A system of imprimitivity f or B over M isa pair h T, P i where: (i) T is a non-degenerate ∗ -representation T of B ; (ii) P is aregular X ( T )-projection-valued Borel measure on M ; and (iii) we have T b P ( W ) = P ( π ( b ) W ) T b (1)for all b ∈ B and all Borel subsets W of G/H . One can show that for any systemof imprimitivity h T, P i h
T, P i ′ φ = Z M dP mT φ ( m ) ( φ ∈ D )is a ∗ -representation of D , and that h T, P i 7→ h
T, P i ′ is a one-to-one correspon-dence between the unitary classes of systems of imprimitivity and unitary classes ofnon-degenerate ∗ -representations of D . For the details and the definition of the in-tegration appearing in (1) we refer the reader to [12, § VIII.18]. In this paper we willalways identify the systems of imprimitivity and non-degenerate ∗ -representations of D . In the rest of this paper, we assume that we have implicitly chosen a transforma-tion group ( G, M ) and use the symbol D to denote the corresponding transformationbundle.For M = G/H with the left translated action, the transformation bundle playsvery important role. Let us provide details. For a B -positive non-degenerate ∗ -representation S of B H , let Y be the Hilbert bundle over G/H constructed asabove, we define a L ( G/H ; Y )-projection valued measure P on G/H by P ( W ) f = Ch W f ( f ∈ L ( G/H ; Y ))(Ch W is the characteristic function) for Borel subsets W ⊂ G/H . Let T = Ind B H ↑ B ( S ).It is easy to show that h T, P i is a system of imprimitivity. If B is saturated, by[12, XI.14.18] the correspondence S
7→ h
T, S i (up to unitary equivalent classes) isactually Rieffel’s inducing process with respect to the pre- C ∗ ( B H )- C ∗ ( D ) Hilbert3imodule L ( B ). We can conclude that C ∗ ( B H ) and C ∗ ( D ) are Morita equivalent.In the case that G is second countable and B is separable, this theorem is proved byKaliszewski, Muhly, Quigg and Williams [15] as an easy consequence of the theoryof equivalence of Fell bundles over locally compact groupoids.For the details on the definition of the regular representation of B we refer thereaders to Exel and Ng [10]. B is said to be amenable if the regular representation,regarded as a representation of the full C ∗ -algebra C ∗ ( B ), is faithful. In the samepaper, the Fell’s Absorption Theorem of the bundle version is proved which wewill use frequently: If T is a non-degenerate ∗ -representation of B such that T | B e is faithful, then inner tensor product T ⊗ R G is weakly equivalent to the regularrepresentation of B , where R G is the left-regular representation of G . In this section, we assume that M = G/H .Let C and C ′ be two Banach bundles with fiber spaces C and C ′ over the samelocally compact space M ′ with Borel measure µ . If F : C → C ′ is a continuous mapsatisfying: (1) F | C x is bounded linear map from C x into C ′ x ; (2) there is constant K > k F | C x k ≤ K for all x ∈ G we say that F is a C - C ′ multiplier oforder e , and we use symbol e F to denote the map from L ( C ) into L ( C ′ ) definedby e F ( f )( x ) = F ( f ( x )) ( f ∈ L ( C ) , x ∈ M ) . Furthermore, if F is norm-preserving and bijective, it is routine to verify that e F isunitary, and we say that F is C - C ′ unitary multiplier of order e . Lemma 2.1.
Let S be a non-degenerate ∗ -representation of B H , U a non-degenerateunitary representation of G . Let h T, P i be the system of imprimitivity induced from S , then the system of imprimitivity induced from the inner tensor product S ⊗ ( U | H ) is unitarily equivalent to ( T ⊗ U, P ⊗ O ( X ( U )) ) .Proof. It is easy to verify that the Hilbert bundle over ( G × G ) / ( H × G ) inducedby the outer tensor product S ⊗ o U is Y ′ = { Y ′ α = Y α ⊗ X ( U ) } α ∈ G/H over
G/H by identifying ( G × G ) / ( H × G ) with G/H . Then we have a unitary operator E : X ( U ) → X ( U ) such that(1 O ( L ( ρ ♯ ; Y )) ⊗ E ∗ )Ind B H × G ↑ B × G ( S ⊗ o U )(1 O ( L ( ρ ♯ ; Y )) ⊗ E )= Ind B H ↑ B ( S ) ⊗ o U. (2)On the other hand, let Z = h Z, Z α i be the Hilbert bundle over G/H induced from S ⊗ ( U | H ), by the proof of [12, XI.13.2] there is a Z - Y ′ unitary multiplier of order e F : Z → Y ′ such that e F Ind ( B H × G ) DH ↑ ( B × G ) DG ( S ⊗ o U | D H ) e F ∗ = Ind B H × G ↑ B × G ( S ⊗ o U ) | D G , (3)4here D G = {h x, x i : x ∈ G } , D H = {h x, x i : x ∈ H } . Let J : Z → Y ′ be definedby J = (1 Y ⊗ E ∗ ) ◦ F , where 1 Y is the identity map from Y to itself, then J isunitary Z - Y ′ multiplier and by (2) and (3) we have e J Ind ( B H × G ) DH ↑ ( B × G ) DG ( S ⊗ o U | D H ) e J ∗ = (Ind B H ↑ B ( S ) ⊗ o U ) | D, hence we have e J Ind B H ↑ B ( S ⊗ ( U | H )) e J ∗ = Ind B H ↑ B ( S ) ⊗ U. Let h Ind B H ↑ B ( S ⊗ ( U | H )) , Q i be the system of imprimitivity induced from S ⊗ ( U | H ),it is easy to see that e J Q e J ∗ = P ⊗ O ( X ) , thus we have e J h Ind B H ↑ B ( S ⊗ ( U | H )) , Q i e J ∗ = h T ⊗ U, P ⊗ O ( X ) i , our proof is done.The following lemma is well-known but we did not find reference. Our proofmight be easier than the standard proof: Lemma 2.2.
Let B be the group bundle C × G , then the ∗ -representation of D which is induced from R H is weakly equivalent to the regular representation of D .Proof. Let h T, P i be the system of imprimitivity induced by R H . Then by Lemma2.1 the system of imprimitivity induced by R H ⊗ ( R G | H ) is h T ⊗ R G , P ⊗ L ( G ) i . Let Q and Q ′ be the integrated forms of h T, P i and h T ⊗ R G , P ⊗ L ( G ) i respectively,then we have Q ′ = Q ⊗ R G . Since Q | D e is faithful, by the Fell’s Absorption Theorem we conclude that Q ′ isweakly equivalent to regular representation of D . On the other hand, by Fell’sAbsorption Theorem again R H ⊗ ( R G | H ) is weakly equivalent to R H , hence Q isweakly equivalent to Q ′ , and so Q is weakly equivalent to regular representation of D , our proof is done.The following lemma is well known, e.g. see Echterhoff and Raeburn [8]. Ourproof based on Lemma 2.1 is easier: Lemma 2.3. R G | H is weakly equivalent to R H .Proof. Let h T, P i be the system of imprimitivity induced by the trivial representa-tion S of H . By Lemma 2.1 the system of imprimitivity induced by S ⊗ ( R G | H )is h T ⊗ R G , P ⊗ O ( L ( G )) i , then by the same argument of the proof of Lemma 2.2 h T ⊗ R G , P ⊗ O ( L ( G )) i is weakly equivalent to the regular representation of D . Thusby Lemma 2.2 S ⊗ ( R G | H ) is weakly equivalent to R H , but S ⊗ ( R G | H ) = R G | H ,the proof is done.The following corollary improve Echterhoff and Quigg [7, Proposition 6.3] inwhich G is discrete and B has AP: Corollary 2.4.
Assume B is saturated and amenable. If for any non-degenerate ∗ -representation of B H we have Ind B H ↑ B ( S ) | H ≥ S , then B H is amenable. In par-ticular, either if H is normal closed subgroup or G/H is discrete, B H is amenable. roof. Let S be a faithful non-degenerate ∗ -representation of C ∗ ( B H ). By Lemma2.3 and [12, XI.13.2]Ind B H ↑ B ( S ⊗ ( R G | H )) | B H = Ind B H ↑ B ( S ) | B H ⊗ ( R G | H )is weakly equivalent to regular representation of B H . On the other hand, since B is amenable, Ind B H ↑ B ( S ⊗ ( R G | H )) is faithful ∗ -representation of C ∗ ( B ), thus wehave Ind B H ↑ B ( S ) | B H ⊗ ( R G | H ) ≥ Ind B H ↑ B ( S ) | B H ≥ S, thus B H is amenable.The other parts of the proof is completed by the combination of [12, XI.14.21]and [12, XI.12.8], which state that either if G/H is discrete or H is normal closedsubgroup then for any non-degenerate ∗ -representation S of B H we have S ≤ Ind B H ↑ B ( S ) | H . Theorem 2.5.
Assume that B is saturated. Let h T, P i be the system of imprimi-tivity induced by S which is non-degenerate ∗ -representation of B H , then h T, P i isweakly equivalent to regular representation of D if and only if S is weakly equivalentto regular representation of B H .Proof. Let h T ′ , P ′ i be faithful ∗ -representation of C ∗ ( D ), S ′ be the ∗ -representationof B H inducing h T ′ , P ′ i , and Q ′ be the integrated form of h T ′ , P ′ i . Then S ′ isfaithful ∗ -representation of C ∗ ( B H ), in particular S ′ | B e is faithful, then by Lemma2.3 S ′ ⊗ ( R G | H ) is weakly equivalent to regular representation of B H . On the otherhand, by Lemma 2.1 h Q ′ ⊗ R G , P ′ ⊗ L ( G ) i is induced by S ′ ⊗ ( R G | H ), and since h Q ′ ⊗ R G , P ′ ⊗ L ( G ) i is weakly equivalent to the regular representation of D , ourproof is completed. C ∗ -Algebraic Bundles and C ∗ -Algebras In this section we study the tensor product of Fell bundles and C ∗ -algebras. Ourmethod is not hard to generalize to construct tensor products of C ∗ -algebras whichwill be treated in a forthcoming paper [14] by the present author. For our goal ofthis paper we confine our attention on this specific case.Let us make some general convention. Let E be a ∗ -algebra. If A is a C ∗ -algebrasuch that there is ∗ -homomorphism r A : E → A such that r ( E ) is norm-dense in A , then we say that A is ∗ - quotient of E , or A is quotient C ∗ - algebra of E . If B is another quotient C ∗ -algebra of E and k r A ( c ) k = k r B ( c ) k for all c ∈ E , thenit is easy to see that r B ( c ) r A ( c ) can be extended to faithful ∗ -homomorphismfrom B onto A . In this case we say that A and B have same ∗ - source , thus if forany C ∗ -algebras A and B we can prove that they have same ∗ -source, then we haveproved that they are ∗ -isomorphic.In this section let A be a fixed C ∗ -algebra, let B x ⊗ A denote the algebraic tensorproduct of B x and A for each x ∈ G . Then we can form a ∗ -algebraic bundle6 d ⊗ A = { B x ⊗ A : x ∈ G } over G by defining( n X i =1 b i ⊗ a i ) ∗ = n X i =1 b ∗ i ⊗ a ∗ i ( b i ∈ B x , x ∈ G ; a i ∈ A ) , ( n X i =1 b i ⊗ a i )( m X j =1 b ′ j ⊗ a ′ j ) = n X i =1 m X j =1 b i b ′ j ⊗ a i a ′ j ( b i ∈ B x , b ′ j ∈ B y , x, y ∈ G ; a i , a ′ j ∈ A . To see these are well defined, we just need toregard { B x ⊗ A : x ∈ G } as subset of algebraic tensor product C ∗ ( B d ) ⊗ A .In the rest of this section we denote the linear span of { x f ( x ) ⊗ a : f ∈ L ( B ) , a ∈ A } by Γ.For any pre- C ∗ -seminorm r of B d ⊗ A we let B x ⊗ r A be the completion of thequotient of B x ⊗ A with respect to the seminorm r | ( B x ⊗ A ). By Lemma 3.2 and[11, II.13.18] we can define a C ∗ -algebraic bundle over G with fibers B x ⊗ r A suchthat all the members of Γ is continuous cross-sections. We denote this C ∗ -algebraicbundle by B ⊗ r A . Notice that if B ⊗ A have pre- C ∗ -seminorms r and r such that r | ( B e ⊗ A ) = r | ( B e ⊗ A ), then r = r and B ⊗ r A = B ⊗ r A .Therefore Γ is norm dense in L ( B ⊗ r A ), and C ∗ ( B ⊗ r A ) is ∗ -quotient of Γ.The proof of the following lemma is routine: Lemma 3.1.
Let T be a non-degenerate ∗ -representation of the ∗ -algebra C ∗ ( B d ) ⊗ A . By regarding { B x ⊗ A } x ∈ G ⊂ C ∗ ( B ) ⊗ A , for each x ∈ G we define semi-norm σ x on B x ⊗ A by T | ( B x ⊗ A ) , then σ = ∪ x ∈ G σ x is a pre- C ∗ seminorm of B ⊗ A . Lemma 3.2.
Let σ be a C ∗ -norm of the ∗ -algebra B e ⊗ A , then there is a pre- C ∗ -seminorm σ ′ on { B x ⊗ A : x ∈ G } such that σ ( c ) ≤ ( σ ′ | B e ⊗ A )( c ) for all c ∈ B e ⊗ A .Proof. Let h S, R i be faithful ∗ -representation of B e ⊗ σ A . We define T = Ind B e ↑ B d (S) . (4)Let Y d be the Hilbert bundle over G d which is induced by S , and let R a be anoperator defined on L ( G d ; Y d ) by R a ( κ x ( b ⊗ ξ )) = κ x ( b ⊗ R a ( ξ )) ( x ∈ G, b ∈ B x , ξ ∈ X ( h S, R i )) . (5)Then R a is in the commuting algebra of T . Furthermore, let F : Y e → X ( S ) be theunitary operator defined by F ( κ e ( b ⊗ ξ )) = S b ξ ( b ∈ B e , ξ ∈ X ( S )) , (6)we have F R a ( κ e ( b ⊗ ξ )) = F ( κ e ( b ⊗ R a ( ξ )))= S b ( R a ( ξ ))= R a S b ( ξ )= R a F ( κ e ( b ⊗ ξ )) (7)7 b ∈ B e , ξ ∈ X ( h S, R i )), hence R = F ∗ ( Y e R ) F. (8)On the other hand, by [12, XI.14.21] F ∗ ( Y e ( T | B e )) F = S, (9)we conclude that if σ ′ is defined by σ ′ ( n X i =1 b i ⊗ a i ) = k n X i =1 T b i R a i k ( n X i =1 b i ⊗ a i ∈ B x ⊗ A, x ∈ G ) , (10)then σ ≤ σ ′ | ( B e ⊗ A ). Finally, by Lemma 3.2 σ ′ is a pre- C ∗ -seminorm on { B x ⊗ A : x ∈ G } .Now we have the following important proposition: Proposition 3.3.
The maximal and minimal norms of B e ⊗ A can be extended tounique pre- C ∗ -seminorms of B d ⊗ A . We denote the corresponding C ∗ -algebraicbundles over G by B ⊗ max A and B ⊗ min A .Proof. If σ is the minimal norm of B e ⊗ A , it is easy to see that the σ ′ constructedin the proof of Lemma 3.2 satisfies σ ′ | ( B e ⊗ A ) = σ . Furthermore, by [12, XI.11.3]if σ is the maximal tensor norm of B e ⊗ A , then σ ′ | ( B e ⊗ A ) = σ .The following lemma is readily proved according to definitions: Lemma 3.4.
The map Ψ σ : n X i =1 b i ⊗ a i n X i =1 b i ⊗ a i ( n X i =1 b i ⊗ a i ∈ B x ⊗ A, x ∈ G ) (11) can be extended to a continuous map from B ⊗ max A onto B ⊗ r A . Therefore, if T is a ∗ -representation of B ⊗ r A , then T ◦ Ψ is a ∗ -representation of B ⊗ max A . Lemma 3.5.
For each b ∈ B x , b i ∈ B y and a i ∈ A ( x, y ∈ G ; i = 1 , ...n ), we have k n X i =1 ( b i b ) ⊗ a i k B ⊗ max A , k n X i =1 ( bb i ) ⊗ a i k B ⊗ max A ≤ k b kk n X i =1 b i ⊗ a i k B ⊗ max A (12) and k n X i =1 ( b i ) ⊗ ( a i a ) k B ⊗ max A , k n X i =1 ( b i ) ⊗ ( aa i ) k B ⊗ max A ≤ k a kk n X i =1 b i ⊗ a i k B ⊗ max A (13) Proof.
We prove k n X i =1 ( b i b ) ⊗ a i k B ⊗ max A ≤ k b kk n X i =1 b i ⊗ a i k B ⊗ max A , (14)8he other parts may be proved by the same argument.By the proof of Lemma 3.2, we have ∗ -representation T of B d and ∗ -representation S of A such that k n X i =1 b i ⊗ a k B ⊗ max A = k n X i =1 T b i S a i k ( n X i =1 b i ⊗ a i ∈ B z ⊗ A, z ∈ G ) . (15)Thus we have k n X i =1 ( b i b ) ⊗ a i k B ⊗ max A = k n X i =1 T b i b S a i k = k T b kk n X i =1 T b i S a i k≤ k b kk n X i =1 b i ⊗ a i k B ⊗ max A . (16)Our proof is done.By the previous lemma, for each b ∈ B and a ∈ A , we can define b u : B ⊗ max A → B ⊗ max A, n X i =1 b i ⊗ a i n X i =1 ( bb i ) ⊗ a i u b : B ⊗ max A → B ⊗ max A, n X i =1 b i ⊗ a i n X i =1 ( b i b ) ⊗ a i ; a v : B ⊗ max A → B ⊗ max A, n X i =1 b i ⊗ a i n X i =1 b i ⊗ ( aa i ) v a : B ⊗ max A → B ⊗ max A, n X i =1 b i ⊗ a i n X i =1 b i ⊗ ( a i a ) . Lemma 3.6. b u , u b , a v and v a are continuous. In particular, h b u, u b i and h a v, v a i are multipliers of B ⊗ max A of order π ( b ) and e respectively.Proof. We prove the continuity of u b , the continuity of the others can be proved bythe same argument.For any f ∈ Γ, it is easy to see that x u b f ( x ) is continuous, for this is theconsequence of the following: If b i → b in B then b i ⊗ a → b ⊗ a in B ⊗ max A . Toprove this, let g ∈ L ( B ) such that g ( π ( b )) = b , then k g ( π ( b i )) − b i k → k g ( π ( b i )) ⊗ a − b i ⊗ a k ≤ k g ( π ( b i )) − b i kk a k → . (17)On the other hand g ( π ( b i )) ⊗ a → g ( π ( b )) ⊗ a = b ⊗ a , by [11, III.13.12] we concludethat b i ⊗ a → b ⊗ a . Therefore we have proved the continuity of x u b f ( x ).Let { c i } i ∈ I ⊂ B ⊗ max A such that c i → c . For arbitrary ǫ >
0, it is easy to seethat there is f ∈ Γ with k f ( π ( c )) − c k < ǫ . Then k f ( π ( c i )) − c i k < ǫ for large i .Thus by Lemma 3.5 we have k u b f ( π ( c i )) − u b c i k < k b k ǫ.
9n the other hand, we have proved that u b f ( π ( c i )) → u b f ( π ( c )) and k u b f ( π ( c )) − u b c k ≤ k b kk f ( π ( c )) − c k < k b k ǫ, by [11, III.13.12] again we have u b c i → u b c , this proved the continuity of u b . Lemma 3.7. b u b and a v a are strongly continuous.Proof. Let { b i } be a net of B converging to b ∈ B . By Lemma 3.6, we have u b i ( n X k =1 b k ⊗ a k ) → n X k =1 bb k ⊗ a k ( b k ∈ B x , x ∈ G ; a k ∈ A ) . Now for any c ∈ B ⊗ max A with π ( b ) = B x , for arbitrary ǫ > b k ∈ B x and a k ∈ A such that k n X i =1 b k ⊗ a k − c k < ǫ. Thus by Lemma 3.5 we have k u b i ( n X i =1 b k ⊗ a k − c ) k < k b i k · ǫ. On the other hand, u b i ( n X k =1 b k ⊗ a k ) → u b ( n X k =1 b k ⊗ a k ) , by [11, III.13.12] we have u b i c → u b c .By Lemma 3.4, Lemma 3.6, Lemma 3.7 and [12, VIII.15.3] we conclude thefollowing proposition: Proposition 3.8.
For any non-degenerate ∗ -representation T of B ⊗ r A , there are ∗ -representations of B and A , say S and R , such that range ( S ) is in the commutingalgebra of R and T ( n X i =1 b i ⊗ a i ) = n X i =1 S b i R a i ( n X i =1 b i ⊗ a i ∈ B x ⊗ A, x ∈ G ) . Proposition 3.9.
Every non-degenerate ∗ -representation of C ∗ ( B ) ⊗ max A is theintegrated form of a unique non-degenerate ∗ -representation of B ⊗ max A .Proof. Let h e S, R i be a ∗ -representation of C ∗ ( B ) ⊗ max A , let S be ∗ -representationof B such that e S is the integrated form of S . It is easy to verify that k n X i =1 S b i R a i k = k n X i =1 b i ⊗ a i k max ( b i ∈ B e , a i ∈ A ) , thus the map T : n X i =1 b i ⊗ a i n X i =1 S b i R a i ( b i ∈ B x , x ∈ G ; a i ∈ A ) , ∗ -representation of ( B ⊗ max A ) d . Furthermore, for each g ∈ Γthe map x T g ( x ) is strongly continuous, thus we conclude that T can be extended to ∗ -representationof B ⊗ max A whose integrated form is h e S, R i .By Proposition 3.9 and Proposition 3.8 together, we conclude that C ∗ ( B ⊗ max A )and C ∗ ( B ) ⊗ max A have same ∗ -source, thus we have proved the first part of thefollowing proposition: Proposition 3.10.
For any C ∗ -algebra A , C ∗ ( B ⊗ max A ) = C ∗ ( B ) ⊗ max A, (18) C ∗ r ( B ⊗ min A ) = C ∗ r ( B ) ⊗ min A Proof.
Let us see the proof of the second part. Let T be faithful ∗ -representationof B , then T | B e is faithful. Let S be a faithful ∗ -representation of A , then T ⊗ S isfaithful representation of the unit fiber of B ⊗ min A , i.e B e ⊗ min A , thusInd B e ⊗ min A ↑ B ⊗ min A (( T | B e ) ⊗ S ) = Ind B e ↑ B ( T | B e ) ⊗ S is weakly equivalent to regular representation of B ⊗ min A . On the other hand,Ind B e ↑ B ( T | B e ) is (weakly equivalent to) regular representation of B and S is faithful ∗ -representation of A , thus C ∗ r ( B ⊗ min A ) and C ∗ r ( B ) ⊗ min A have the same ∗ -sourceΓ, they are ∗ -isomorphic. Corollary 3.11. If C ∗ ( B ) is nuclear, then B e is nuclear.Proof. Let A be a C ∗ -algebra, and σ any C ∗ -norm of B ⊗ A . Let S and R benon-degenerate faithful ∗ -representation of B and A respectively, and T be faithful ∗ -representation of B ⊗ σ A . Then Ind B e ⊗ σ A ↑ B ⊗ t ( σ ) , which is ∗ -representation of C ∗ ( B ) ⊗ max A by Proposition 3.8, is weakly contained in S ⊗ R because C ∗ ( B ) isnuclear. Thus by [11, XI.11.3] we have T ≤ Ind B e ⊗ σ A ( T ) ≤ ( S ⊗ R ) | ( B e ⊗ σ A )= S | B e ⊗ R, so σ | ( B e ⊗ A ) is equivalent to the minimal C ∗ -norm of B e ⊗ A , this proved that B e is nuclear.Combine Corollary 3.11 and Proposition 3.10, we have: Proposition 3.12.
Let B be an amenable C ∗ -algebraic bundle over G . Then the C ∗ -algebra C ∗ ( B ) = C ∗ r ( B ) is nuclear if and only if: (i) B e is nuclear; (ii) for any C ∗ -algebra A , B ⊗ min A is amenable. Approximation Property of C ∗ -Algebraic Bun-dles Recall from Exel and Ng [10], for any α, β ∈ L ( G, B e ) we can define a map Φ α,β : B → B by Φ α,β ( b ) = α · b · β = Z G α ( x ) ∗ bβ ( π ( b ) − x ) dx ∈ B π ( b ) ( b ∈ B )and by [10, Lemma 3.2] Φ induces a map Ψ α,β : L ( B ) → L ( B ) defined byΨ α,β ( f )( y ) = Z G α ( x ) ∗ f ( y ) β ( y − x ) dx = Φ α,β ( f ( y )) ( f ∈ L ( B )) . Definition 4.1. B is said to have AP (i.e approximation property) if there is M > and nets { α i } , { β i } ⊂ L ( G, B e ) such that: (i) sup i k α i kk β i k ≤ M ; (ii) Ψ α i ,β i ( b ) → b uniformly on compact slices of B ( [10, Definition 3.6] ). Let T be any non-degenerate ∗ -representation of B such that r = T | B e is faithful,let µ T = T ⊗ R G . For each γ ∈ L ( G, B e ) we define V Tγ : X ( T ) → L ( G, X ( T )) by V Tγ ( ξ ) = r ( γ ( x ))( ξ ) ( ξ ∈ X ( T )) . By the proof of [10, Lemma 3.1], for any α, β ∈ L ( G, B e ) we have e T (Ψ α,β ( f ))( ξ ) = ( V Tα ) ∗ f µ T ( f ) V Tβ ( ξ ) ( ξ ∈ X ( T )) . (19)This motivated us to give the following definition: Definition 4.2.
We say that B have Ultra-Approximation Property (UAP) if thereis a net { Ψ i } of maps Ψ i : L ( B ) → L ( B ) such that that for any non-degenerate ∗ -representation T of B there are nets { V i } i ∈ I , { W i } i ∈ I ⊂ O ( X ( T ) , L ( G )) satisfythe following:i. we have e T (Ψ i ( f )) = W ∗ i f µ T ( f ) V i ( f ∈ L ( B )) . (20) Furthermore, if R ∈ O ( X ( T )) is in the commuting algebra of T we have V i R = ( R ⊗ O ( L ( G )) ) V i ; (21) ii. there is constant M > such that k V i k , k W i k ≤ M for all i ;iii. For any f ∈ L ( B ) , Ψ i ( f ) → f in the norm of C ∗ ( B ) .If these conditions hold, we say that { W i } i ∈ I , { V i } i ∈ I are nets of { Ψ i } i ∈ I under T . It is easy to prove that if B has UAP then B is amenable, and by (19) if B hasAP it has UAP. 12 emark 4.3. To construct UAP, we usually define a map in a dense subset of L ( B ) and then extend it. In order to accomplish this, we need the following generaleasy observation: Let B be an arbitrary C ∗ -algebraic bundle over G , and T be afaithful ∗ -representation of C ∗ ( B ) . Then by [12, VIII.16.4] we can identify each f ∈ L ( B ) with T f . Now let M be a norm-dense subset of L ( B ) , F : M → L ( B ) bea map. If F : T ( L ( B )) → O ( X ( T )) is a continuous map such that F ( T f ) = T F ( f ) for each f ∈ M , then we can conclude that F ( T ( L ( B ))) ⊂ T ( L ( B )) , thus F is a map from L ( B ) into L ( B ) as an extension of F , furthermore it is easy toverify that F is continuous with respect to the norm of L ( B ) . Proposition 4.4. If B has UAP, then D has UAP, in particular it is amenable.Proof. Let { Φ i } i ∈ I be UAP of B . Let Γ be the linear span of the cross-sections withthe form x f ( x ) r for some f ∈ L ( B ) and r ∈ C ( M ). Define Φ ′ : Γ → L ( D )by Φ ′ i ( f r ) = Φ i ( f ) r f ∈ L ( B ) , r ∈ C ( M ) . Let h T, R i be a faithful ∗ -representation of D , and { V i } i ∈ I and { W i } i ∈ I be thenets of { Φ i } under T . Define φ : h T, R i ( L ( D ) → O ( X ( h T, R i ))by φ ( h T, R i ( f r )) = ( W i ) ∗ µ h T,R i ( f r ) V i ( f ∈ L ( B ) , r ∈ C ( M )) . (22)By Remark 4.3 if we can prove that φ ( h T, R i (Φ ′ i ( f r )) = T (Φ i ( f )) R r ( f ∈ L ( B ) , r ∈ C ( M )) (23)then each Φ ′ i is extendable to a map from L ( B ) into L ( B ). (23) is derived bythe following: φ ( h T, R i ( f r )) = ( W i ) ∗ µ h T,R i ( f r ) V i = ( W i ) ∗ µ T ( f ) R r ⊗ O ( L ( G )) V i = ( W i ) ∗ µ T ( f ) V i R r = T (Φ i ( f )) R r . Now let us verify that Φ ′ i satisfies 4.2(i). Let h T, R i be arbitrary ∗ -representation of B ⊗ r A , we have h T, R i (Φ ′ i ( f r )) = T (Φ i ( f )) R r = ( W i ) ∗ µ T ( f ) V i R r = ( W i ) ∗ µ T ( f ) R a r ⊗ O ( L ( G )) V i = ( W i ) ∗ µ h T,R i ( f r ) V i for all f ∈ L ( B ) and r ∈ C ( M ), by the linearity and continuity of Φ ′ i , (20) isproved. Furthermore, notice that any operator which is in the commuting algebraof h T, R i is in the commuting algebra of T , (21) holds. The verification of 4.2(ii)and4.2(iii) are routine, we omit them. 13 roposition 4.5. If B has UAP, then B ⊗ r A has UAP, in particular it is amenable.Proof. Let { Φ i } i ∈ I be UAP of B . Let Γ be the linear span of the cross-sections withthe form x f ( x ) ⊗ a for some f ∈ L ( B ) and a ∈ A . Define Φ ′ i : Γ → L ( B ⊗ r A )by Φ ′ i ( f ⊗ a ) = Φ i ( f ) ⊗ a f ∈ L ( B ) , a ∈ A. By the same argument of Proposition 4.4 we can prove that each Φ ′ i can be extendedto a map from L ( B ⊗ r A ) to itself, which we still denote by Φ ′ i , such that { Φ ′ i } i ∈ I is the UAP of B ⊗ r A . Remark 4.6.
In Proposition 4.5, if B has AP, it is hard to check whether B ⊗ r A has AP because we do not know how to identify compact slices in B ⊗ r A in order tocheck 4.1 (ii). The same difficulty occurs in the study of the transformation bundle,and even worst it is hard to see how to define the AP net of D according to the APnet of B . In this sense, UAP is a more economic concept than AP.Combine Proposition 4.4 and Proposition 4.5, we can construct many “non-trivial” examples of C ∗ -algebraic bundles which have UAP. For instance, let ( G, M ) be an amenable G -transformation group, by Claire Anantharaman-Delaroche [4,Lemma 2.4] it is easy to verify that the semi-direct product bundle ( C ( M ) , G ) hasAP, thus has UAP. Now for any C ∗ -algebra A and anothoer transformation group ( G, M ′ ) , we can form tensor product of ( C ( M ) , G ) and A , and furthermore thetransformation bundle over G derived from this tensor product, all of them haveUAP, so they are all amenable. But it is difficult to check whether they have AP. The following is our main theorem:
Theorem 4.7.
Let B be a saturated C ∗ -algebraic bundle over G with UAP (inparticular if B has AP). Then for any closed subgroup H ⊂ G the restriction bundle B H is amenable, and C ∗ ( B H ) = C ∗ r ( B H ) is nuclear if and only if B e is nuclear.Proof. By Theorem 2.5 and Proposition 4.4 B H is amenable. The proof of theother part is the combination of Theorem 2.5, Proposition 3.12, Proposition 4.4 andProposition 4.5.The ‘if’ part of the following corollary is well-known in varied specific forms: Corollary 4.8. If G is amenable locally compact group and B is a saturated C ∗ -algebraic bundle over G , then C ∗ ( B ) = C ∗ r ( B ) is nuclear if and only if B e is nuclear. Bibliography [1] Fernando Abadie, Alcides Buss, and Damin Ferraro,
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