Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras

In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.


Introduction
The notion of a Schur multiplier has its origins in the work of I. Schur in the early 20 th century, and is based on the entrywise (or Hadamard) product of matrices. More specifically, a bounded function ϕ : N × N → C is called a Schur multiplier if (ϕ(i, j)a i,j ) is the matrix of a bounded linear operator on 2 whenever (a i,j ) is such. Hence to a Schur multiplier ϕ : N × N → C we can associate an operator S ϕ on B( 2 ). Based on a concrete description of Schur multipliers which was given by A. Grothenieck in [8], Schur multipliers can be identified with ∞ ⊗ eh ∞ , the extended Haagerup tensor product of two copies of ∞ .
Among the large number of applications of Schur multipliers is the description of the space M A cb (G) of completely bounded multipliers, also known as Herz-Schur multipliers of the Fourier algebra A(G) of a locally compact group G, introduced by J. de Canniere and U. Haagerup in [3]. Namely, as shown by M. Bozejko and G. Fendler [1], M cb A (G) can be isometrically identified with the space of all Schur multipliers on G × G of Toeplitz type. Furthermore, in connection with Schur multiplier, Herz-Schur multiplier has very important application in the study of the nuclearity of the group C * -algebra, i.e for a discrete group G, the reduced group C * -algebra C * r (G) is nuclear if and only if there is a net of completely positive Herz-Schur multiplier ϕ i : G → C such that ϕ i (x) → 1 for all x ∈ G (see e.g [2]).
Recently, in [10], McKee, Todorov and Turowska generalized the notion of Schur multipliers and Herz-Schur multipliers to the C * -algebra valued case: a class of Schur A-multipliers and a class of Herz-Schur multiplier of semidirect product bundle are identified, where A is a C *algebra faithfully represented on a Hilbert space H. In this 'operator-valued' case, the starting point is a function φ defined on the direct product X × Y , where X and In this paper, we will generalize the notion of Herz-Schur multiplier to Fell bundle B over locally compact group by borrowing the ideas of [10] and [9], then we will give a necessary and sufficient condition of that C * r (B) is nuclear C * -algebra in terms of multipliers when G is a discrete group, finally as an application we prove: If B is a Fell bundle over discrete group G such that C * r (B) is nuclear, then for any subgroup H ⊂ G the C * -algebra C * r (B) is nuclear as well. The plan of this paper is: In the Section 2, we give some notations and conventions which we will use in this paper; in the Section 3, we define Schur multipliers of Fell bundle; in Section 4, we give a characterization of Schur multipliers of Fell bundle which is analogous to [10,Theorem 2.6], and during this process we include the non-unital version Stinpring's Theorem in a proposition which will be very important in Section 6 but with an easier proof; in Section 5 we study the generalization of Herz-Schur multipliers in the context of Fell bundles, including the generalized transference theorem between Schur multipliers and Herz-Schur multipliers (see e.g [10,Theorem 3.8]); finally in Section 6 we study the problem concerning the nuclearity of the reduced C * -algebra by aid of the notion of the generalized Herz-Schur multipliers.

Preliminary
We refer the reader to [6,II.13] for the notion of Banach bundles, [7, VIII.2, VIII.3, VIII.16] for the notion of C *algebraic bundles, and [7,VIII.9] for the basic knowledge about the representation theory of C * -algebraic bundles. As usual, we call C * -algebraic bundles Fell bundles.
In this section we give a brief review of some basic definitions and knowledge which we will use in the rest of this paper. The definitions of notations are given in the paragraphs labeled as 'Remark'. When we use some notations we will refer to the corresponding remarks in order to remind the reader.
Remark 2.1. If D is a Banach bundle over a locally compact Hausdorff space M with a fixed Borel measure, we use the symbol L(D) to denote the space of continuous cross-sections vanishing outside some compact subset of M , and the symbol L p (D) to denote the space of the p-integrable cross-sections of D (see [6,II.15]). If D is a trivial Banach bundle with constant fiber A (see [6,II.13.6]), we write L (D) (resp. L p (D)) as L (M, A) (resp. L p (M, A)).
Remark 2.2. Recall that, by [6,II.15.2], if X is a Hilbert space, then L 2 (M, X) is a Hilbert space with the inner product where ( , ) is the inner product of X.
The following lemma is trivial, we list it here for reference: Lemma 2.1. Let M and N be two locally compact Hausdorff spaces, φ : M → N a continuous map, and D a Banach bundle over N with bundle space B . Let C be the Banach bundle over M which is the retraction of D by φ : M → N , whose bundle space we denote by Z (see [6,II.13.7]).
is a cross-section of C. Furthermore, f is continuous if and only if f is continuous. We call f the cross-section of C by canonical identification of f . Remark 2.3. In the rest of this paper, we will identify f and its canonical identification f , i.e, if f : M → B is a (continuous) map such that f (m) ∈ B φ(m) , we will regard f as a (continuous) cross-section of C.
Remark 2.4. Throughout this paper, we assume that G is a fixed group which is either discrete with counting measure or is locally compact and second countable with a fixed Haar measure, and we use the symbol µ to denote either the counting measure or the Haar measure according to whether G is discrete or not. Furthermore, B = B, π, ·, * is a fixed Fell bundle over G. If G is not discrete, we assume that each fiber space B x is separable.
We refer the reader to [7,VIII.5] to see how to make L 1 (B) as a Banach * -algebra.We use the symbol C * (B) to denote the C * -completion (see [6,VI.10]) of L 1 (B).
Remark 2.5. We use the symbol B×B to denote the Banach bundle over G × G which is the retraction from B by the continuous map φ : G × G → G, (x, y) → xy −1 , and we denote its bundle space by D. For any continuous map Remark 2.7. Assume that X is a Hilbert space. We denote the space of all bounded linear operators of X by O(X), and the space of all compact operators by O c (X). If τ is a representation of a Fell bundle B or a C * -algebra A on X, sometimes we write X as X(τ ) in order to emphasis that the representation τ is acting on X.
Remark 2.8. If τ is a * -representation of B, we use the symbol τ int to denote the integrated form of τ (see [7,VIII.11]). Remark 2.9. If A is a C * -algebra, we denote its multiplier algebra by M (A). Recall that if A is a concrete C *algebra acting on a Hilbert space X, then M (A) ⊂ O(X). Furthermore, if S : A → O(Y ) is a non-degenerate *representation of A on a Hilbert space Y , then S can be uniquely extended to M (A). Therefore, we can regard S as a non-degenerate * -representation of M (A) on Y .
Definition 2.1. Recall that, by definition, the bundle space D of B × B is a (topological) subspace and map * : Since the multiplication and involution are continuous in B, we conclude that and * are continuous.
Furthermore, since ρ : B → O(X(ρ)) is strong operator continuous, for any k ∈ L(B × B) the map (s, t) → ρ D (k(s, t)) is a strong-operator continuous map from G × G into O(X(ρ)).
Recall that the bundle space B of B can be identified with a subset of M (C * (B)) such that the topology of B is stronger than the relativized topology of B inherited from the strict topology of M (C * (B)) (see [7] and [5]). Thus, for a ∈ B and b ∈ C * (B), we have multiplications ab and ba in M (C * (B)) such that ab ∈ C * (B) and ba ∈ C * (B). We define : D × C * (B) → C * (B) by (s, t; a) b = ab ((s, t; a) ∈ D, b ∈ C * (B)).
For the sake of convenience, we use the same symbol to denote the map from C * (B) × D to C * (B) defined by b (s, t; a) = ba ((s, t; a) ∈ D, b ∈ C * (B)).
By definitions we have and similarly we have We state the following simple lemma without proof: Remark 2.10. Let Z be any set, and W ⊂ Z a subset. We use the symbol χ W to denote the character function of W , i.e., χ W is defined by Remark 2.11. Our final remark is about the integration theory of Banach bundles. Let C be an arbitrary Banach bundle over a locally compact Hausdorff space M with Borel measure ν. We use the symbol R(C) to denote the set of the linear span of {χ W k : k ∈ L(C) and W is compact subset of M }, (see Remark 2.1 for the notations) where χ W is the character function of W , and χ W k is the cross-section of C defined by χ W k (x) = χ W (x)k (x) (x ∈ M ). By [6, Chapter II], for any k ∈ L p (C) (see Remark 2.1 for the notations) there is a sequence {k n } n∈N ⊂ R(C) such that for almost x ∈ M . In particular, since k n (x) − k(x) p ≤ ( k n (x) + k(x) ) p for almost x ∈ M and that the function x → ( k n (x) + k(x) ) p is integrable, by applying Lebesgue's Dominated Theorem, we have

Schur Multipliers of Fell Bundles
In this section we define the notion of Schur multipliers of Fell bundles. We assume that ρ is a * -representation on a Hilbert space X, and sometimes we will write X as X(ρ) (see Remark 2.7). Recall that D is the bundle space of B × B (see Remark 2.5).
Proposition 3.1. Let k ∈ L 2 (B × B) (see Remark 2.1 for the notations). For arbitrary ξ ∈ L 2 (G, X) (see Remark 2.2), the map defined by P x k,ξ : G → X, y → ρ D (k(x, y))(ξ(y)) (see Definition 2.2) is integrable for almost x ∈ G, and the map . Therefore, we can associate an operator T ρ k on the Hilbert space L 2 (G, X) defined by We have T ρ k ≤ k 2 . proof. Let k ∈ R(B × B) (see Remark 2.11 for the notations). Assume that ξ ∈ L 2 (G, X) having the form of ξ = n i=1 χ Vi ξ i (see Remark 2.10 for the notation), where V i ⊂ G are compact and ξ i ∈ X. Then it is easy to verify that P x k,ξ is integrable for almost x ∈ G. Furthermore, by Fubini's Theorem, Q k,ξ is measurable and is compactly supported. Combining with we conclude that Q k,ξ is in L 2 (G, X). Now we assume that k is an arbitrary element of L 2 (B×B), and ξ an arbitrary element of L 2 (G, X). We prove that P x k,ξ is integrable for almost x ∈ G. Let {ξ n } n∈N be a sequence of simple functions of L 2 (G, X) such that ξ n (x) − ξ(x) → 0 for almost x ∈ G, and ξ n − ξ 2 → 0; on the other hand, let for almost all y ∈ G. Furthermore, there is a null set L ⊂ G such that for all n ∈ N and x ∈ G \ L, the map y → ρ D (k n (x, y))(ξ n (y)) is measurable and vanishes outside compact subsets. Combining these statements, we conclude that if x ∈ G \ (M ∪ L), the map y → ρ D (k(x, y))(ξ(y)) is measurable and vanishing outside countable union of compact This proves that P x k,ξ is integrable for almost

By our definitions of the null subsets
. This proves that Q k,ξ is measurable, and it is clear that Q k,ξ is vanishing outside a countable union of compact subsets of G. Furthermore, by the same argument which derived (10), we have We conclude that Q k,ξ is in L 2 (G, X).
The other parts of this proposition is easy consequence of our previous discussion.
Remark 3.1. We use the symbol E(ρ, B) to denote the norm- (9), Remark 2.7 and Remark 2.2 for the notations).
(see Definition 2.1 for the notations) is continuous and compactly supported for all (x, y) ∈ G × G, and the map proof. Since k 1 and k 2 are continuous cross-sections, the is a continuous from G×G×G into D, where c ki : G×G → B is the map satisfying k i (x, y) = (x, y; c ki (x, y)) (i = 1, 2) (see Remark2.1), and, since k i (i = 1, 2) is compactly supported, so is p. Furthermore, by the continuity of p and the equality we conclude that the map z → k 1 (x, z) k 2 (z, y) is continuous. In order to complete our proof, since it is sufficient to prove that (recall that by Lemma 3.1, k 1 k 2 ∈ L(B × B)), and the symbol k * 1 to denote the cross-section defined by The following lemma can be verified by routine computation, we omit its proof: Lemma 3.2. For any k 1 , k 2 ∈ L(B × B), we have (see (9) and Definition 3.1 for the notations) Therefore, E(ρ, B) (see Remark 3.1 for the notations) is a C * -algebra.
Remark 3.2. We denote the set of all simple functions l ∈ L 2 (G × G, C * (B)) (see Remark 2.1 for the notations) having the form of Since we assumed that ρ is a non-degenerate *representation, the concrete Remark 2.9). In Lemma 3.3 below, we will prove that Remark 3.3. Let us recall from [10] that, for any k ∈ ) for almost (x, y) ∈ G × G, and the maps k l and l k defined by (see (7) and (8) for the notations) are in L 2 (G × G, C * (B)). Furthermore we have (see Proposition 3.1 and Remark 3.3 for the notations) In particular, since {T ρint proof. Let l be a function with the form of (14). Suppose supp(k) ⊂ E × F , where E and F are compact subsets of G. Our first task is to prove that the map z → k(x, z) l(z, y) Then q is bounded, measurable and compactly supported by For any fixed x and y, we define p x,y : G → C * (B) by p x,y (z) = q(x, z, y)(= k(x, z) l(z, y)). Then, by the fact that we proved q ∈ L 1 (G × G × G, C * (B)) and Fubini's Theorem, the map k l : (x, y) → G p x,y (z)dz is measurable. On the other hand, since q is compactly supported by E ×( and bounded, we conclude that k l : (x, y) → G p x,y (z)dz = G q(x, z, y)dz is bounded, measurable and compactly supported by E × n i=1 F i . Therefore, we conclude that k l ∈ L 2 (G × G, C * (B)).
By our previous discussion, for almost (x, y) for almost x ∈ G. This proves that T ρint Then the following map (see Remark 3.3 for the notations) . Therefore, we can regard Ξ ρ,r as a * -homomorphism from M (O c (L 2 (G)) ⊗ ρ int (B)) onto M (O c (L 2 (G)) ⊗ r int (B)) (see Remark 2.9) satisfying (see (9) for the notation) and proof. Since r is weakly contained in ρ, the map S : For each k ∈ L(B × B) and l ∈ K(B) (see Remark 3.2 for the notations), by Lemma 3.3, we have Since {T ρint l : l ∈ K(B)} is dense in O c (L 2 (G)) ⊗ r int (C * (B)) and S ⊗ I L2(G) is non-degenerate, (18) implies that Now (16) is proved for k ∈ L(B × B). But since L(B) is dense in L 2 (B), (16) holds for k ∈ L 2 (B × B).
(17) is a direct consequence of Lemma 3.3. The proof is complete. Proposition 3.2. Let k ∈ L 2 (B × B). Then T ρ k =0 if and only if k(x, y) = 0 for almost all (x, y) ∈ G × G.
proof. The 'if' part is trivial, we prove the 'only if' part. Let R : ρ int (C * (B)) → O(Y ) be a non-degenerate faithful *representation of ρ int (C * (B)) on a separable Hilbert space Y, and r : B → O(Y ) the * -representation of B on Y such that R • ρ int is the integrated form of r. Since ρ and r are weakly equivalent, the map Ξ ρ,r , defined in Lemma 3.4, is faithful *homomorhpism, hence T ρ k = 0 implies that T r k = 0. Since ρ int (C * (B)) is a separable C * -algebra, by [10], T r k = 0 if and only if k(x, y) = 0 for almost all (x, y) ∈ G × G. Our proof is complete.
Definition 3.2. Let D be a Banach bundle over locally compact space M with bundle space D . We call a continuous map Φ : Φ·k , is completely bounded linear map, then we say that Φ is a Schur (B, ρ)-multiplier. Fuarthermore, if S ρ , Φ is completely positive, we say Φ is completely positive Schur (B, ρ)multiplier.
If Φ is a Schur (B, ρ)-multiplier, we define Therefore, if Φ is a Schur (B, ρ)-multiplier, S ρ,Φ can be extended to a completely bounded map from E(ρ, B) (see Remark 3.1 for the notations) into O(L 2 (G, X(ρ))). Thus we will always consider that S ρ,Φ is a completely bounded map defined on E(ρ, B).
The following proposition is useful in our further study: is a non-degenerate *representation of B on a Hilbert space Y which is weakly equivalent to ρ, then Φ is a Schur (B, r)-multiplier if and only if it is a (B, ρ)-multiplier.

Characterization of Schur (B, ρ)-multipliers
Our main result of this section is the characterization of Schur (B, ρ)multipliers (See Definition 3.3). In order to achieve this goal, we define a specific class of completely bounded maps in Definition 4.2.
In the rest of this section, let A be a fixed C * -algebra, M (A) the multiplier algebra of A (see Remark 2.9), and X a fixed Hilbert space. Recall that O(X) is the space of all bounded linear operators of X.
is said to be a representation of f . If R is non-degenerate, we say that Let C ⊂ M (A) be a C * -subalgebra. By a completely bounded (S C,A )-extendable map from C into O(X) or briefly a (S C,A )-extendable map on C, we shall mean a completely bounded map g : C → O(X) which has an extension on M (A) which is (S A )-map. We say that such extension is an (S C,A )-extension of g.
It is easy to see that if r :

Let us define r : M (A) → O(Z) by r(a) = Er (a)E
for all a ∈ M (A) (recall that E is in the commuting algebra of r because Z is r-stable), then r is a non-degenerate *representation of M (A) such that r| A is non-degenerate. Let {a i } i∈I ⊂ A be a norm-bounded net such that a i → a strictly, then r(a i ) → r(a) in strong * -operator topology of O(Z) provided i → ∞. Hence we have If we define W = EW and V = EV , (21) implies that Therefore, if we set Y = Z, then (W, V, r, Y ) is a representation of f such that r(A) acts on Z non-degenerately.
Conversely, suppose that f has a non-degenerate representation (W, V, r, Y ) such that r| A is non-degenerate * -representation of A on Y . Let {a i } i∈I be a norm-bounded net of elements of A such that a i → a in M (A) strictly, we have On the other hand, by the same argument we have f (a) * (ξ) = lim i→∞ f (a i ) * (ξ) (ξ ∈ X).
Combining these two qualities, we conclude that f (a i ) → f (a) in strong * topology of O(X). Our proof is complete.
provided that a ∈ M (A) and {a i } i∈I ⊂ A is a norm-bounded net converging to a strictly, and the right side limit is in strong * -topology of O(X). (ii) f has a representation (V, R) (see the second paragraph of Definition 4.1); (iii) let {a i } i∈I be any approximate unit of A which satisfies a i ≤ 1, then we have It implies that is a completely bounded map, then f is a (S C,A )-extendable map. Furthermore, if f is completely positive, the extension of f can be chosen to be completely positive.
proof. Since we can extend f to a completely bounded map from A into O(Y ), then by Corollary 4.1, we conclude that f has a (S C,A )-extension.
Now suppose that f is completely positive. By Corollary 4.2, we have a non-degenerate representation (V, r) of f . Let us extend r to a non-degenerate * -representation r of the C *algebra C = {b + t1 M (A) : b ∈ C, t ∈ C} − · M (A) (to see the existence of the extension, notice that C is a closed * -ideal of C ). We define g : C → O(X) by Then g is completely positive. Since C is a C * -subalgebra In the rest of this section, we will apply our definitions and propositions to C * -algebras A = O c (L 2 (G)) ⊗ ρ int (C * (B)) and C = E(ρ, B) (see Lemma 3.2 and Lemma 3.3). Recall that if G is non-discrete, both A and C are separable C *algebra by Remark 2.6.
Proposition 4.1. Let us denote O c (L 2 (G))⊗ρ int (C * (B)) = A and E(ρ, B) = C (see Lemma 3.2). Recall that, from Lemma 3.3, C ⊂ M (A). Let f : C → O(L 2 (G, X(ρ))) (See Remark 2.7) be a completely bounded map. Consider the conditions: (i) f is (S C,A )-extendable; (ii) f has a non-degenerate representation (W, V, R) such that for some * -representation r of B which is weakly contained in ρ.
Then we have (ii) ⇒ (i), and if either G is discrete or It is clear that r • ρ int : C * (B) → O(Z) is non-degenerate, thus r • ρ int is the integrated form of a * -representation of B on Z which we denote by r. Then r is weakly contained in ρ. For fixed k ∈ L(B × B), by Lemma 3.3, we have On the other hand, by Lemma 3.3 again, we have Since K(B) is dense in A and S| A is non-degenerate *representation of A, we conclude that Now let W = U W , V = U V , and define a *homomorphism R : E(ρ, B) → E(r, B) by R(T ρ k ) = T r k (k ∈ L 2 (B × B)) (see Lemma 3.4).Since {T ρ k : k ∈ L 2 (B × B)} is dense in E(ρ, B), (25) implies that E(ρ, B)).
By the similar argument, we can prove that (i) ⇒ (ii) if G is discrete.
(ii) ⇒ (i): Since r is weekly contained in ρ, then by Lemma 3.4, the following map Since S is non-degenerate, by Lemma 4.1, F is a (S A )-map on M (A) and an extension of f . Our proof is complete.
For any a ∈ L ∞ (G), we define a ∈ L ∞ (G) by a(s) = a(s) for all s ∈ G. We associate an operator M a : It is clear that M * a = M a , thus M(G) = {M a : a ∈ L ∞ (G)} is a concrete C * -algebra acting on the Hilbert space for a ∈ L ∞ (G) and s, t ∈ G. Then it is easy to verify that a·k, k · a are in L 2 (B × B). Furthermore, for any k ∈ L 2 (B × B) and a, b ∈ L ∞ (G), If Φ is a Schur (B, ρ)-multiplier, we have for all x ∈ E(ρ, B) and a, b ∈ L ∞ (G). Theorem 4.1. Let us denote O c (L 2 (G))⊗ρ int (C * (B)) = A and E(ρ, B) = C. Recall that, from Lemma 3.3, C ⊂ M (A). If Φ : D → D be a multiplier of B × B, and consider the following two statements: (ii) There is non-degenerate * -representation r : B → O(Y ) of B on a Hilbert space Y (or denoted by Y (r)) which is weakly contained in ρ, and such that ρ(Φ((x, y; a)) = W * (x)r D ((x, y; a))V (y) ((x, y; a) ∈ D x,y ) (see Definition 2.2 for the notations) for almost all (x, y) ∈ G × G. Then we have (ii) ⇒ (i), and if either X(ρ) is separable or G is discrete then we have (i) ⇒ (ii).
(If G is discrete, then by [7,VIII.16.11], for any *representation ρ of B we always have ρ(B) ⊂ ρ int (C * (B)). Then we have Corollary 4.3 implies that every completely bounded multipliers is (S C,A )-extendable. Therefore, we can remove 'S ρ,Φ is (S C,A )-extendable' in (i), and (i) and (ii) are always equivalent.) proof. We prove that (i) ⇒ (ii) if X(ρ) is separable, the argument in the case that G is discrete is similar. Since S ρ,Φ is (S C,A )-extendable, by Proposition 4.1 there are separable Hilbert space Y , non-degenerate * -representation r : B → O(Y ) which is weakly contained in ρ, and bounded operators 2 (B × B)).
For any d ∈ L ∞ (G) and k ∈ L 2 (B × B), we have and Let W = EW 0 , combining (28) and (27), we have for all k ∈ L 2 (B) and ξ, η ∈ L 2 (G, X). We conclude that Furthermore, since C is stable under the action of the C * - We conclude that By the same argument we can replace V 0 by an operator V ∈ L ∞ (G, O(X, Y )). By the same argument of the proof of [10, Theorem 2.6], we have, for any k ∈ L 2 (B × B), for almost all (x, y) ∈ G × G.
Since B × B is second-countable Banach bundle over second countable locally compact space G × G, L(B × B) is separable. Let {k n } n∈N ⊂ L(B × B) be a countable dense subset of L(B × B). In particular, for fixed x, y ∈ G, the set {k n (x, y) : n ∈ N} is dense in D x,y . On the other hand, for each n ∈ N we define n (x, y)))}.
Then N = n∈N N n is a null-subset of G × G. If (x, y) ∈ (G×G)\N , then for any a x,y ∈ D x,y , since there is a sequence {k ni } i∈N ⊂ {k n } n∈N such that k ni (x, y) → a x,y , we have The proof of (i) ⇒ (ii) is complete. Hence, Since r is weakly contained in ρ, by Proposition 4.1, our proof is complete.

Herz-Schur Multipliers
In this section, we fix ρ : B → O(X) to be a non-degenerate * -representation of B. We remind the reader that ρ int is the integrated form of ρ.
Let Ψ : B → B be a multiplier of B (see Definition 3.2). For each f ∈ L(B), we define Ψ · f ∈ L(B) by is completely bounded. In this case, S ς Ψ can be extended to a completely bounded map on which we still denote by S ς Ψ . Remark 5.1. Notice that if r and ς are weakly equivalent *representation of B, then it is easy to see that Ψ is (ς, B)multiplier if and only if Ψ is (r, B)-multiplier, and in this case We use the symbol λ B to denote the regular * -representation of C * (B), and we denote the reduced C * -algebra of B, i.e C * (B)/Ker((λ B ) int ), by C * Red (B). For the details about regular * -representationand the reduced C * -algebra, we refer the reader to [5].
Let λ : G → O(L 2 (G)) be the left regular representation of G. By [7,VIII.9.16], we can form a * -representation ρ ⊗ λ of B on L 2 (G) ⊗ X(ρ), which is weakly equivalent to λ B (by [5]), defined by From [5] we can identify C * Red (B) with Recall that, for any f ∈ L(B) and ξ ∈ L 2 (G, X), we have for almost all x ∈ G. Definition 5.2. We call a multiplier Ψ : is completely positive, we call Ψ completely positive Herz-Schur multiplier of B.
If Ψ is Herz-Schur multiplier, we define Remark 5.2. By Remark 5.1, Definition 5.2 is independent of the choice of * -representation of ρ : B → O(X) satisfying that ρ| Be is faithful.
Therefore, if Ψ is a Herz-Schur multiplier, S ρ⊗λ is completely bounded map defined on C * Red (B). In the following, if Ψ is a Herz-Schur multiplier, we will denote S ρ⊗λ Ψ briefly by S Ψ .
is a completely bounded map, let (W, V, R, Z) be its representation.
Therefore, let r : B → O(Z) be the * -representation of B whose integrated form is R • ς int , we have Now let y ∈ G and a y ∈ B y , and {g i } i∈I ⊂ L(G) a net such that supp(g i ) → y, g i ≥ 0 and G g i (x)dx = 1.
On the other hand, Therefore, we have ς(Ψ(a y )) = W * r(a y )V (y ∈ G; a y ∈ B y ) (ii) ⇒ (i) : Since r is weakly contained in ς, then we have a * -homomorphism R : ς int (C * (B)) → r int (C * (B)) satisfying Therefore, for arbitrary ξ ∈ Y , we have It is routine to check that N (Ψ) satisfies (i)-(ii) of Definition 3.2, thus N (Ψ) is a multiplier of B × B.
If either (i) or (ii) holds, we have proof. (i) ⇒ (ii) : By Proposition 5.1, we have a * - By Proposition 4.1 and the same argument of (i) ⇒ (ii) of Proposition 4.1, it is not hard to prove that N (Ψ) is Schur Hence, (ρ ⊗ λ) int (Ψ · f ) cb = S N (Ψ) cb . The proof is complete.
is a * -representation of B which weakly contains λ B , then any (ς, B)-multiplier is Herz-Schur multiplier.
proof. This is the combination of Theorem 5.1, Proposition 5.1 and Theorem 4.1.

Nuclearifty of Cross-Sectional Algebra
In this section we assume that B is a Fell-bundle over a discrete group G, and ρ a fixed * -representation of B which weakly contains λ B (the regular * -representation of B). Recall that, under these assumptions, we have Therefore, by Lemma 4.3 and Theorem 4.1 we have Corollary 6.1. Let Φ : D → D be a multiplier of B × B, then the following are equivalent: (i) Φ is (resp. completely positive) Schur (B, ρ)-multiplier.
Remark 6.1. Let us recall that each D x,x = B e (x ∈ G). However it is not necessary that all Φ ρ x are identical. Proposition 6.1. Let Φ : D → D be a completely positive Schur (ρ, B)-multiplier, then each Φ ρ x is completely positive. Furthermore, proof. By Corollary 6.1, each Φ ρ x is completely positive. Let {a i } i∈I be an approximate unit of the C * -algebra B e with a i ≤ 1 (i ∈ I). Let A be the collection of all finite subsets of G. We define the order on A × I by We define a U,i = a i for all (U, i) ∈ A × I. It is clear that {a U,i } (U,i)∈A×I is an approximate unit of B e since it is a subnet of {a i } i∈I . By [7,VIII.5.11] and [7, VIII.16.3], we have a U,i b − b → 0 for all b ∈ C * (B).
For any multiplier Ψ : B → B of B, we define Ψ ρ x : ρ(B x ) → ρ(B x ) by Ψ ρ x (ρ(a x )) = ρ(Ψ(a x )) (a x ∈ B x ) for all x ∈ G. Definition 6.1. We call a Fell bundle B nuclear if there exists a net {Ψ} i∈I of completely positive Herz-Schur multipliers of B such that i. Ψ ρ e cb ≤ 1 for all i ∈ I; ii. Each (Ψ i ) ρ x has finite dimensional range (x ∈ G; i ∈ I); iii. (Ψ i ) ρ x (ρ(a x )) − ρ(a x ) → 0 provided i → ∞ for all s ∈ G and a x ∈ B x .
Assume that A is nuclear C * -algebra. Recall that there is a net {φ i } i∈I of completely positive contractive maps on A such that the range of each φ i is finite and φ i (a) − a → 0 for all a ∈ A provided i → ∞. We call {φ i } i∈I an approximation net of A.
Furthermore, by 6.2, we have h F H.S = (h F ) e ≤ F cb .
Theorem 6.1. The following are equivalent: (i) B is nuclear; (ii) C * Red (B) is nuclear C * -algebra. If either of this hold, we have C * (B) = C * Red (B). proof. By the aid of Proposition 6.2, we can prove the equivalence of (i) and (ii) by the argument of [9,Theorem 4.3]. If (ii) holds, then by [4,Theorem 25.11] C * (B) = C * Red (B). Corollary 6.3. If B is a Fell bundle over discrete group G such that either C * (B) or C * Red (B) is nuclear C * -algebra, then for any subgroup H ⊂ G, C * (B H ) and C * Red (B H ) is nuclear C * -algebra. Furthermore, B H is amenable.
proof. Notice that if C * (B) is nuclear, then C * Red (B) is nuclear because the quotient C * -algebra of a nuclear C *algebra is nuclear.
If C * Red (B) is nuclear, then by Theorem 6.1 we have a