Coactions on Operator spaces and Exactness Chi-Keung Ng Abstract In this paper, we define and study coactions on operator spaces (which can be regarded as comodules of Hopf C ∗ -algebras) and coactions on operator algebras as well as their crossed products. In particular, we prove that operator exact sequence of operator spaces are preserved under crossed products of coaction by coamenable Hopf C ∗ -algebras.
0. Introduction and Notations The aims of this paper are to give a precise definition of actions (in the case of locally compact groups) and coactions (in the case of Hopf C ∗ -algebras) on operator spaces and to study briefly the properties of this structure. In fact, coactions on operator spaces can be regarded as comodules of a given Hopf C ∗ -algebras. If we want to define comodule structure (of that Hopf C ∗ -algebra) on a Banach space, we shall first come across the problem of finding a suitable norm on the tensor product for the range of the coproduct. There are also some other technical difficulties. However, by considering operator spaces instead of Banach spaces, these problems can be solved. Hence operator spaces is a natural candidate for the comodules of Hopf C ∗ -algebras to be defined. This is one reason why we study coactions on operator spaces. Moreover, we shall see in the second section that any “non-degenerate corepresentation” of a Hopf C ∗ -algebra define a coaction on the underlying column Hilbert spaces. In the case of a locally compact group G, there is a one to one correspondence between bounded representations of G on a Hilbert space and coactions (or equivalently, completely bounded actions) of C0 (G) on the column operator space of that Hilbert space (see Example 2.3 (b) & (c)). A similar result holds for Hopf von Neumann algebras (see Proposition 2.11). This makes the notion of coactions even more interesting. Throughout this paper, we shall view operator spaces as a special kind of Banach K-bimodules (Proposition 1.2 and Lemma 1.3) using a form of Ruan’s characterisation. This identification simplified some of the proofs in this paper. Moreover, this makes the subject look more algebraical. In order to define coactions on operator spaces, we need to study the operator space structure on the multipliers of operator bimodules. We shall do this in Section 1 (after recalling some basic materials on operator spaces). In Section 2, we shall define coactions on operator spaces by Hopf C ∗ -algebras and give a bunch of examples. All of these examples are very natural. However, since some of them are a bit technical, we shall present them as propositions. Note that even the notion of completely bounded actions of locally compact groups on operator spaces (or operator algebras) seems to be a new subject even though it is a very natural thing to consider. Next, we shall define, as in the case of C ∗ -dynamical systems, the reduced crossed product of coactions on operator spaces (as well as those on operator algebras) by Hopf C ∗ -algebras. Note that these reduced crossed products are right Hopf bimodules (see the end of Section 3) of the dual Hopf
1
C ∗ -algebras. We shall give a duality type theorem for reduced crossed products. The idea of the proof is similar to the case of C ∗ -algebras (see [2]). In the final section, we shall investigate the relation between operator exact sequences of operator spaces and their reduced crossed products. In particular, we want to answer the following natural question: given an operator exact sequence, q
i
0→X→Y →Z→0 (in the sense that i is a complete topological injection and q is a complete topological surjection such that ker(q) = i(X)), and coactions X , Y and Z respectively on X, Y and Z which are compatible with i and q; Is the reduced crossed product, q×id i×id 0 → X ×X ,r Sˆ −→ Y ×Y ,r Sˆ −→ Z ×Z ,r Sˆ → 0,
also an operator exact sequence? Note that in the case of C ∗ -algebras, this holds for full crossed products instead of the reduced ones. Therefore, we need to assume that the underlying multiplicative unitary V is amenable. One difficulty in proving this general case is that the corresponding proof in the case of C ∗ -algebras depends very much on the algebraic structure. This difficulty can be overcome by regarding the reduced crossed products as operator SˆV -bimodules instead of just operator spaces and then use the results about operator space structures on their multipliers (see Section 1 and Section 3). The above result can be used to show the following: If V is a “nice” amenable multiplicative unitary and E is an operator exact (see Definition 4.1) operator algebra or operator space with coaction by SV , then the reduced crossed product is also operator exact. This is a generalisation of the result in [15]. Note that the operator exactness defined in this paper is a stable version of the exactness defined in [20] (in particular, they coincide with each other in the case of stable operator spaces; see Remark 4.7). Throughout this paper, unless specified, X, Y and Z are operator spaces, K and B are respectively the sets of compact operators and bounded operators on l2 . Moreover, A is a C ∗ -algebra and (S, δ) is ˆ a Hopf C ∗ -algebra. We shall use ⊗ to denote the spatial tensor product of operator spaces while ⊗ means the operator projective tensor product and is the algebraic tensor product. Furthermore, we denote x · a the module scalar multiplication while yb is the multiplication inside a C ∗ -algebra.
I. Operator spaces and operator bimodules We begin this section by recalling the notion of operator spaces. Note that we use a similar approach as [13] and consider operator spaces as a kind of “Banach L∞ -K-bimodules”. Definition 1.1: Let B be a normed ∗-algebra and N be a normed space. (a) A norm k · kα on B N is said to be a module cross norm if it is a cross norm such that for any a, b ∈ B and z ∈ B N , ka · z · bkα ≤ kakB kzkα kbkB (where a · (c ⊗ n) · b = acb ⊗ n). (b) A module cross norm is said to be L∞ if for any disjoint (self-adjoint) projections p, q ∈ B (i.e. pq = 0), kp · z · p + q · z · qkα = max{kp · z · pkα , kq · z · qkα }. Proposition 1.2: (a) For any Banach space N , there is an one to one correspondence between the operator space structures on N and the L∞ -module cross norms on K N . In this case, the L∞ -module cross norms on K N is given by the spatial operator tensor product K ⊗ N . (b) A linear map S from X to Y is completely bounded if and only if the map idK ⊗ S extends to a bounded map from K ⊗ X to K ⊗ Y . In this case, kSkcb = kidK ⊗ Sk. 2
(c) There exists an injective map from K CB(X; Y ) to CB(X; K ⊗ Y ) which gives the natural operator space structure on CB(X; Y ). The following proposition is probably well known. Here, LK means the set of all bounded K-bimodule maps. Lemma 1.3: LK (K ⊗ X; K ⊗ Y ) = {idK ⊗ S : S ∈ CB(X; Y )}. Consequently, CB(X; Y ) ∼ = LK (K ⊗ X; K ⊗ Y ) as normed spaces. Definition 1.4: A linear map S from X to Y is said to be a complete topological surjection (respectively, complete topological injection or complete metric surjection) if idK ⊗ S is a topological surjection (respectively, topological injection or metric surjection) from K ⊗ X to K ⊗ Y . Let X be a closed subspace of an operator space Y . There is a canonical operator space structure on Y /X (see e.g. [21]). One way to describe this structure is to consider the bidual of Y /X. More precisely, as the annihilators of X, denoted by X ⊥ , is an operator subspace of Y ∗ , we can define an operator space structure on Y /X by regarding it as a subspace of (X ⊥ )∗ . Hence (Y /X)∗ = X ⊥ as operator space. Throughout this paper, we shall assume this operator space structure on Y /X. Remark 1.5: (a) If X is an operator subspace of Y , X ∗∗ can be regarded as the operator subspace X ⊥⊥ of Y ∗∗ . Since (Y /X)∗∗ (∼ = (X ⊥ )∗ ) can be regarded a subspace of (X ⊥ )∗∗∗ and Y ∗∗ /X ⊥⊥ is a ⊥⊥ ⊥ ∗ subspace of ((X ) ) . (Y /X)∗∗ ∼ = Y ∗∗ /X ⊥⊥ completely isometrically (note that as X ⊥ is a subspace ∗ ⊥ ⊥⊥ ∼ ⊥ ∗∗ of Y , (X ) = (X ) by the above argument). (b) The canonical quotient map from Y to Y /X is completely bounded (since X ⊥ is an operator subspace of Y ∗ , the map from Y ∗∗ to (Y /X)∗∗ is completely bounded). (c) If ϕ is a completely bounded map from Y to Z and X ⊆ Ker(ϕ), then the induced map ϕˆ from Y /X to Z is also completely bounded. In fact, ϕ∗ (Z ∗ ) ⊆ X ⊥ and this induces a completely bounded map ψ from (Y /X)∗∗ to Z ∗∗ . It is clear that the restriction of this map on Y /X is ϕ. ˆ In this case, kϕk ˆ cb ≤ kψkcb ≤ kϕ∗ kcb ≤ kϕkcb . Lemma 1.6: If T is a completely bounded map from Y to Z such that T ∗ is a complete isometry, then T is a complete metric surjection. Consequently, if X is a closed subspace of Y , then the canonical quotient map q from Y to Y /X is a complete metric surjection. ˆ ∗ (by [9, 4.1]) and (B∗ ⊗Y ˆ ∗ )∗ = CB(Y ∗ ; B). Since T ∗ is a Proof: We first note that (K ⊗ Y )∗ = B∗ ⊗Y ∗∗ complete isometry and B is injective, (idK ⊗ T ) is a metric surjection from CB(Y ∗ ; B) to CB(Z ∗ ; B). Hence idK ⊗ T is also a metric surjection. In the remainder of this section, we shall study operator space structure on multipliers of operator bimodules. We first recall the following notion. Let N be a Banach A-bimodule. A linear map l (respectively, r) from A to N is said to be a left (respectively, right) multiplier of N if l(ab) = l(a) · b (respectively, r(ab) = a · r(b)) for any a, b ∈ A. We denote by MAl (N ) (respectively, MAr (N )) the set of all left (respectively, right) multipliers of N . Furthermore, (l, r) is called a multiplier of N if l ∈ MAl (N ) and r ∈ MAr (N ) such that a · l(b) = r(a) · b for any a, b ∈ A. Let MA (N ) be the collection of all multipliers. As in the literatures, we say that a Banach A-bimodule N is essential if both A · N and N · A are dense in N . Lemma 1.7: Let A be a C ∗ -algebra and N be an essential A-bimodule. (a) Any left or right multiplier on N is automatically bounded. (b) For any (l, r) ∈ MA (N ), klkMAl (N ) = krkMAr (N ) . (c) MA (N ) is a Banach space. Proof: The proof for part (a) is exactly the same as that for [19, 3.12.2]. To show part (b), we first note that by [18, 5.2.2], A has a contractive approximate identity for N . It is then easy to see that 3
klkMAl (N ) = sup{ka · l(b)kN : kak ≤ 1; kbk ≤ 1} = sup{kr(a) · bkN : kak ≤ 1; kbk ≤ 1} = krkMAr (N ) . Part (c) follows easily from part (a). Suppose that Y is an operator A-bimodule in the sense that Y is a Banach A-bimodule and K ⊗ Y is a Banach K ⊗ A-bimodule. Suppose that Y is also essential. Then it is not hard to see that K ⊗ Y is an essential K ⊗ A-bimodule. From now on, unless specified, all operator bimodules are essential. Suppose that Y is a closed subspace of L(H; K) (where H and K are Hilbert spaces) and ψ and φ are faithful non-degenerate representations of A on L(H) and L(K) respectively such that φ(A)Y ψ(A) ⊆ Y . Then Y is called a spatial operator A-bimodule. It is natural to ask whether all operator bimodules are spatial. Indeed, it is true for the essential ones (note that on the contrary, a spatial operator A-bimodule need not be essential). Lemma 1.8: Let Y be an essential operator A-bimodule. Then there exist Hilbert spaces H and K as well as complete isometry π from Y to L(H; K) and faithful non-degenerate representations ψ and φ of A on L(H) and L(K) respectively such that φ(b)π(y)ψ(a) = π(b · y · a) for all a, b ∈ A and y ∈ Y . Proof: By [8, 3.3], there exist a Hilbert space L as well as complete isometry π 0 from Y to L(L) and injective (but not necessarily non-degenerate) ∗-homomorphisms ψ 0 and φ0 from A to L(L) such that φ0 (b)π 0 (y)ψ 0 (a) = π 0 (b · y · a) for any a, b ∈ A and y ∈ Y . Let H = ψ 0 (A)L and K = φ0 (A)L and let ψ and φ be respectively the restrictions of ψ 0 and φ0 on H and K. Then clearly, φ and ψ are faithful nondegenerate representations of A. Since Y is essential, for any x ∈ Y and ξ ∈ L, π 0 (x)ξ = φ0 (b)π 0 (y)ψ 0 (a)ξ for some y ∈ Y and a, b ∈ A. Hence π 0 (Y )H ⊆ K and let π be the restriction of π 0 on H. It remains to show that π is a complete isometry. In fact, it is clear that kπ(x)k ≤ kπ 0 (x)k = kxk. On the other hand, for any > 0, there exists a, b ∈ A such that kak, kbk ≤ 1 and kb · x · a − xk < . Hence kxk < sup{kφ0 (b)π 0 (x)ψ 0 (a)ξk : ξ ∈ L; kξk ≤ 1} + ≤ kπ(x)k + and π is an isometry. Now if we replace X by K ⊗ X and A by K ⊗ A, we have (1 ⊗ ψ 0 (A))(l2 ⊗ L) = l2 ⊗ H and (1 ⊗ φ0 (A))(l2 ⊗ L) = l2 ⊗ K. Consequently, idK ⊗ π from K ⊗ X to L(l2 ⊗ H; l2 ⊗ K) is an isometry and π is a complete isometry. ˜ A (Y ) = {m ∈ L(H; K) : φ(A)m, mψ(A) ⊆ Now there is an alternative way to define multipliers: M π(Y )}. However, it is not obvious that this definition is independent of the choice of representations. Nevertheless, we shall see later that it is indeed completely isometrically isomorphism to MA (Y ) (as operator A-bimodules). Let us first define a natural operator space structure on MA (Y ). By Lemma 1.7(a) and Proposition 1.2(b), any left or right multiplier from A to Y is automatically completely bounded (consider e.g. idK ⊗ l and idK ⊗ r). Hence we can consider an operator space structure on MAl (Y ) by regarding it as a closed subspace of CB(A; Y ). In this case, the operator space structure is given by the canonical injection from K MAl (Y ) to MAl (K ⊗ Y ) ⊆ CB(A; K ⊗ Y ) (see Proposition 1.2(c)). By default, we shall consider the norm k · k on MAl (Y ) induced by CB(A; Y ) and for simplicity, we shall use k · kusu to denote the norm k · kMAl (Y ) (or k · kMAr (Y ) ) on MAl (Y ) (respectively, MAr (Y )) induced by L(A; Y ) (note that k·kusu ≤ k·k). Since CB(A; Y ) ∼ = LK (K⊗A; K⊗Y ) (Lemma 1.3), l (K ⊗Y ), k·kusu ) is an isometry. Therefore, the operator the canonical map from (MAl (Y ), k·k) to (MK⊗A l space structure on MAl (Y ) is given by the canonical embedding K MAl (Y ) to (MK⊗A (K⊗K⊗Y ), k·kusu ). l Now Lemma 1.7(b) implies that (MK⊗A (K ⊗ Y ), k · kusu ) is a norm subspace of both (MK⊗A (K ⊗ r l Y ), k · kusu ) and (MK⊗A (K ⊗ Y ), k · kusu ). Thus, the norms induced on MA (Y ) by (MA (Y ), k · k) and (MAr (Y ), k · k) coincide. Moreover, by replacing Y with K ⊗ Y , we obtain a norm on K MA (Y ) which gives a natural operator space structure on MA (Y ).
Proposition 1.9: Let Y be an essential operator A-bimodule. Then there exists a completely isometric ˜ A (Y ) such that Ψ(l, r)a = l(a) and aΨ(l, r) = r(a) for any (l, r) ∈ isomorphism Ψ from MA (Y ) to M MA (Y ) and a ∈ A. Moreover, k · kusu and k · k coincide on MA (Y ). Proof: Let φ, π and ψ be any spatial realisation of (Y, A) (see Lemma 1.8) and {ai } be an approximate unit of A. Suppose that (l, r) is an element in MA (Y ). Then π(l(ai )) will converge strongly to an 4
element m ∈ L(H; K) (as ψ is non-degenerate and l is bounded). It is clear that mψ(a) = π(l(a)) and φ(b)mψ(a) = π(b · l(a)) = π(r(b))ψ(a) for a, b ∈ A. Moreover, kmk = sup{kmψ(a)k : a ∈ A; kak ≤ 1} = klkusu = k(l, r)kusu . It is not hard to see that the map Ψ that sends (l, r) to m is ˜ A (Y ) (note that any element in M ˜ A (Y ) defines an a surjective isometry from (MA (Y ), k · kusu ) to M element in MA (Y )) which satisfies the required equalities. It remains to show that k · kusu coincides with k · k and Ψ is a complete isometry. Observe that by replacing Y with K ⊗ Y and A with K ⊗ A, ˜ K⊗A (K ⊗ Y ). For any k ∈ K and a ∈ A, we have an isometry Ψ0 from (MK⊗A (K ⊗ Y ), k · kusu ) to M ˜ A (Y ) (recall Ψ0 (id⊗l, id⊗r)(k⊗a) = (1⊗Ψ(l, r))(k⊗a). Hence Ψ is an isometry from (MA (Y ), k·k) to M that k(l, r)k = k(idK ⊗ l, idK ⊗ r)kusu ). This shows that k · k and k · kusu agree on MA (Y ). Now, if we ˜ A (K⊗Y ). For any k, k 0 ∈ K and replace Y by K⊗Y , we have an isometry Φ from (MA (K⊗Y ), k·k) to M a ∈ A, k⊗(l, r) can be regarded as an element of MA (K⊗Y ) and Φ(k⊗(l, r))(k 0 ⊗a) = (k⊗Ψ(l, r))(k 0 ⊗a). ˜ A (K ⊗ Y ) is an isometry and hence Ψ is a complete isometry Therefore, idK ⊗ Ψ from K ⊗ MA (Y ) to M by Proposition 1.2(b). From now on, we may use the above identification implicitly. ˜ A (Y ). This applies, Remark 1.10: If Y in the above proposition is an operator algebra, then so is M in particular, to the case when Y = B ⊗ A for some operator algebra B. Corollary 1.11: (a) Y is an operator subspace of MA (Y ). (b) MA (Y ) is a unital operator M (A)-bimodule. (c) If B is another C ∗ -algebra and Z is an essential operator B-bimodule, then there exists a complete isometry from MB (MA (Y )⊗Z) to MA⊗B (Y ⊗Z) that respects both the A-bimodule and the B-bimodule structures. From now on, we shall regard MB (MA (Y ) ⊗ Z) as subspace of MA⊗B (Y ⊗ Z). Lemma 1.12: Let X and Y be essential operator A-bimodules. Suppose that ϕ is a completely bounded A-bilinear map from X to Y . Then ϕ induces a completely bounded M (A)-bimodule map ϕ˜ from MA (X) to MA (Y ). Moreover, if ϕ is completely isometric, then so is ϕ. ˜ Proof: Let ϕ be the completely bounded map from CB(A; X) to CB(A; Y ) induced by ϕ. For any (l, r) ∈ MA (X), it is clear that ϕ(l, ˜ r) = (ϕ(l), ϕ(r)) is in MA (Y ) and this defines a completely bounded map from MA (X) to MA (Y ). It is easy to check that ϕ˜ is a M (A)-bimodule map. Finally, if ϕ is a complete isometry, so is ϕ¯ and hence ϕ˜ is a complete isometry. For simplicity, we shall again use ϕ (instead of ϕ) ˜ to denote the extension of ϕ on MA (X). Lemma 1.13: Let A and B be C ∗ -algebras and ψ be a non-degenerate ∗-homomorphism from A to M (B). (a) idX ⊗ψ extends to a complete contraction, also denoted by idX ⊗ψ, from MA (X ⊗A) to MB (X ⊗B) such that (idX ⊗ ψ)(m · a) = (idX ⊗ ψ)(m) · ψ(a) and (idX ⊗ ψ)(a · m) = ψ(a) · (idX ⊗ ψ)(m) for any m ∈ MA (X ⊗ A) and a ∈ M (A). Moreover, if ψ is injective, then idX ⊗ ψ is a complete isometry. (b) Let Y and Z be an essential operator A-bimodule and an essential operator B-bimodule respectively. Suppose that T1 and T2 are linear maps from MA (Y ) to MB (Z) such that T1 (y) = T2 (y) (for all y ∈ Y ) and either Ti (l(a)) = Ti (l, r)·ψ(a) (for any a ∈ A, (l, r) ∈ MA (Y ) and i = 1, 2) or Ti (r(a)) = ψ(a)·Ti (l, r) (for any a ∈ A, (l, r) ∈ MA (Y ) and i = 1, 2). Then T1 = T2 . Proof: Since part (b) is obvious, we need only to show part (a). Consider a complete isometry from X to L(H), then by Proposition 1.9, MA (X ⊗ A) can be regarded as a subspace of M (K ⊗ A). It is not hard to see that (id ⊗ ψ)(MA (X ⊗ A)) is a subspace of MB (X ⊗ B) ⊆ M (K ⊗ B). This gives the required map. The last statement of part (a) is clear. Remark 1.14: (a) If T : X → Y is a completely bounded map and ψ : A → M (B) is a non-degenerate ∗-homomorphism, then (T ⊗ idB ) ◦ (idX ⊗ ψ) = (idY ⊗ ψ) ◦ (T ⊗ idA ) (as maps from MA (X ⊗ A) to MB (Y ⊗ B)). 5
(b) By considering the non-degenerate ∗-homomorphism from A to M (A ⊗ B) that send a ∈ A to a ⊗ 1, we see that the map j from MA (X ⊗ A to MA⊗B (X ⊗ A ⊗ B) given by j(˜ x) = x ˜ ⊗ 1 is a well defined A-bilinear complete isometry. Note that the complete isometry in part (b) will be used implicitly throughout the whole paper. We end this section with the following result about quotient bimodule. The first part of the following result is probably well known while the second part follows from the above consideration. Lemma 1.15: Let Y be an operator A-bimodule and X be a closed submodule of Y . Then Y /X is also an operator A-bimodule. If, in addition, Y is an essential operator A-bimodule, then the quotient map extends to a completely bounded M (A)-bimodule map from MA (Y ) to MA (Y /X).
II. Coactions on operator spaces We can now define coactions on operator spaces. Definition 2.1: (a) Let be a completely bounded map from X to MS (X ⊗ S). Then is said to be a coaction of S on X if ( ⊗ id) = (id ⊗ δ) ∈ CB(X; MS⊗S (X ⊗ S ⊗ S)). Moreover, a coaction is said to be right non-degenerate if (X) · S (the linear span of {(x) · s : x ∈ X; s ∈ S}) is norm dense in X ⊗ S. Similarly, we can define left non-degenerate as well as (2-sided) non-degenerate coactions. (b) If B is an operator algebra, then a coaction of S on B is a coaction in the sense of part (a) together with the extra assumption that is an algebraic homomorphism from B to MS (B ⊗ S) (see Remark 1.10(a)). (c) Let X and Y be coactions of S on X and Y respectively. Then a completely bounded map T from X to Y is said to be equivariant if (T ⊗ idS ) ◦ X = Y ◦ T . Remark 2.2: The coaction identity in Definition 2.1(a) actually means that Φ ◦ ( ⊗ id) ◦ = (id ⊗ δ) ◦ (where Φ is the forgettable complete isometry from MS (MS (X ⊗ S) ⊗ S) to MS⊗S (X ⊗ S ⊗ S) given by Corollary 1.11(c)). Example 2.3: (a) Let Γ be a discrete group. If is an injective coaction of L Cr∗ (Γ) on a C ∗ -algebra A, then is non-degenerate (by [1, 7.15]) and A can be decomposed as A = r∈Γ Ar (by [14, 2.6]). L Let F be any subset of Γ and AF = r∈F Ar . Then the restriction F of on AF is a coaction of Cr∗ (G) on AF . Moreover, it is not hard to see that this coaction is also non-degenerate. In particular, if Λ is a subsemigroup of Γ, then Λ is a coaction of Cr∗ (Γ) on the operator algebra AΛ (in the sense of Definition 2.1(b)). On the other hand, suppose that is a right (or left) non-degenerate coaction of Cr∗ (Γ) on an operator space X. Let Xr = {x ∈ X : (x) = x ⊗ λr }. Then Xr = (id ⊗ ϕr )(X) where ϕr is the functional on Cr∗ (Γ) such that ϕr (λt ) = δr,t as defined in [14, §2] (δr,t means the Kronecker L delta). Now by the right (respectively, left) non-degeneracy of , X = r∈Γ Xr . (b) Let G be a locally compact group. Then coactions of C0 (G) on X are in one to one correspondence with completely bounded actions of G on X in the following sense: an action α of G on X is said to be completely bounded if (i) there is λ > 0 such that sup{k(idK ⊗ αt )(¯ x)k : t ∈ G} ≤ λk¯ xk for any x ¯ ∈ K X, (ii) for any x ∈ X, α• (x) is a continuous map from G to X. Moreover, if the coaction is a complete isometry, then condition (i) is replaced by the following condition: (i)’ sup{k(idK ⊗ αt )(¯ x)k : t ∈ G} = k¯ xk for any x ¯ ∈ K X. In fact, by considering X as a subspace of a C ∗ -algebra, we see that MC0 (G) (X ⊗ C0 (G)) = Cb (G; X). Hence as in the case of C ∗ -algebras, δ induces an action α of G on X such that αt (x) = δ(x)(t). Since δ(X) ⊆ Cb (G; X) and δ is bounded, there exists λ0 > 0 such that sup{kαt (x)k : t ∈ G} ≤ λ0 kxk and for fixed x ∈ X, α• (x) is a continuous map from G to X. As δ is completely bounded, we can replace X by 6
K ⊗ X and hence α is a completely bounded action. Conversely, let α be a completely bounded action. If we define for any x ∈ X and t ∈ G, δ(x)(t) = αt (x), then condition (ii) implies that δ(x) ∈ C(G; X) (i.e. a continuous map) and so k ⊗ δ(x) ∈ C(G; K ⊗ X) (for any k ∈ K). Furthermore, condition (i) shows that id ⊗ δ is a bounded map from K X to Cb (G; K ⊗ X) = MC0 (G) (K ⊗ X ⊗ C0 (G)). It is not hard to check that δ is a coaction in the sense of Definition 2.1(a) and the required correspondence is established. Next, we investigate the situation when δ is injective. In fact, δ is injective if and only if αt is injective for all (and equivalently, for some) t ∈ G (as αs αt = αst ). It is the case if and only if αe = IX (note that αe (αe (x) − x) = 0). Furthermore, if X is an operator algebra, then δ is an algebraic homomorphism if and only if all αt (t ∈ G) are algebraic homomorphisms. (c) There is a one to one correspondence between completely bounded actions and completely bounded representations. A map T from G to CB(X; X) is called a completely bounded representation if (i) Tr ◦ Ts = Trs ; (ii) sup{kTr kcb : r ∈ G} < ∞; (iii) for any x ∈ X, T• (x) is a continuous map from G to X. In particular, if H is a Hilbert space and Hc is the column operator space of H, then coactions of C0 (G) on Hc are exactly bounded continuous representations of G on L(H) = CB(Hc ; Hc ). Hence there is a one to one correspondence between bounded continuous representations of G on L(H) and coactions of C0 (G) on Hc . The following example is expected. However, since the definition of coactions in [1] requires quite a lot of identifications (in particular, (idE ⊗δ)(T ) = T ⊗idA ⊗δ 1 and (δE ⊗id)(T ) = (V ⊗1S )(T ⊗δA ⊗idS 1)), it is not obvious whether the coaction identity in [1] yields the coaction identity we want. Proposition and Example 2.4: Let E be a Hilbert A-module and (δE , δA ) be a coaction of S on (E, A) as defined in [1, 2.2]. Then δE defines a completely isometric coaction E on the column operator space Ec . Proof: Let KE = K(E ⊕ A). We define, as in [1, 2.6], the isometry iE from L(A; E) to L(E ⊕ A) by iE (t)(y ⊕ a) = t(a) ⊕ 0 and regard E as a subspace of L(A; E) in the obvious way (in this case, iE (E) ⊆ KE ). This gives operator space structures on L(A; E) and on E. The structure on E is the ˜ (E ⊗ S)c column operator space structure (see e.g. [4]). We shall show that MS (Ec ⊗ S) equals M ˜ (where M (E ⊗ S) = {F ∈ L(A ⊗ S; E ⊗ S) : (1E ⊗ S)F, F (1A ⊗ S) ⊆ E ⊗ S} as in [1, 2.1]). In fact, by Lemma 1.12, iE ⊗ id is a complete isometry from MS (Ec ⊗ S) to MS (KE ⊗ S) ⊆ M (KE ⊗ S). Its ˜ (E ⊗ S) −→ L((E ⊗ S) ⊕ (A ⊗ S)) ∼ image coincides with the image of the map jE : M = M (KE ⊗ S) induced by iE⊗S (see [1, 2.6]). Hence we can define E : Ec −→ MS (Ec ⊗ S) by (iE ⊗ id) ◦ E = jE ◦ δE .
(1)
[1, 2.7] shows that δE is a complete isometry under the operator space structures defined above and hence so is E . It remains to show the coaction identity for E . First of all, we shall show that (iE ⊗ idS⊗S ) ◦ Φ ◦ (E ⊗ id) ◦ E = (δKE ⊗ id) ◦ δKE ◦ iE
(2)
(where Φ is the map as described in Remark 2.2 and δKE is the coaction defined in [1, 2.7]). Because of the definition of Φ, it is not hard to see that (iE ⊗ idS⊗S ) ◦ Φ = ((iE ⊗ id) ⊗ id) (if we identify MS (M (KE ⊗ S) ⊗ S) as a subspace of M (KE ⊗ S ⊗ S)). Thus, (2) is obtained if we can show that ((iE ⊗ id) ⊗ id) ◦ (E ⊗ id) = (δKE ⊗ id) ◦ (iE ⊗ id) (since δKE ◦ iE = (iE ⊗ id) ◦ E by [1, 2.7] and (1)). In fact, it is clear that this equation holds on E ⊗ S and so the equation holds on MS (E ⊗ S) (by Lemma 1.13(b)). On the other hand, we would like to prove that (iE ⊗ idS⊗S ) ◦ (id ⊗ δ) ◦ E = (id ⊗ δ) ◦ δKE ◦ iE . Again, it suffices to show that (iE ⊗ idS⊗S ) ◦ (id ⊗ δ) = (id ⊗ δ) ◦ (iE ⊗ id) and this equation clearly follows from Lemma 1.13(b). Now as δKE is a coaction and (idE ⊗ idS⊗S ) is a complete isometry, we obtain the coaction identity for E . 7
Example 2.5: In the case of a Hilbert space H, a coaction on H (in the sense of [1]) is the same as a unitary corepresentation on H. In fact, consider A = C with a trivial coaction δ. Then H ⊗δ S = H ⊗ S and we have, from [1, 2.3], a unitary V ∈ L(H ⊗ S). It is not hard to see that the first condition of [1, 2.3] automatically holds while the second condition is equivalent to the condition for corepresentation. Hence unitary corepresentation of S on H gives rise to a coaction of S on Hc . More generally, we have a partial generalisation of Example 2.3(c) concerning the relation between non-degenerate corepresentations of S and coactions. Let us first give some definitions. For any Hilbert space K, an element W in M (K(K) ⊗ S) ⊆ L(K ⊗ S) (not necessary a unitary) is said to be a corepresentation of S if (id ⊗ δ)(W ) = W12 W13 . A corepresentation is said to be non-degenerate if there exists an approximate unit {ei } in S and a dense subspace K 0 of K such that for any s ∈ S and ξ ∈ K 0 , the limit of {(1 ⊗ s)(W (ξ ⊗ ei ))} exists in K ⊗ S. It is clear that if S is unital (i.e. a compact quantum group) or commutative (i.e. a locally compact semigroup), then any corepresentation is automatically non-degenerate. Moreover, we have the following lemma which shows that the non-degeneracy of W follows from the non-degeneracy of the induced representation πW of S ∗ (when S = SˆV for some regular multiplicative unitary V ). Lemma 2.6: Let V be a regular multiplicative unitary on H and W a corepresentation of SˆV on K. If the induced representation πW of SˆV∗ is non-degenerate, then so is W . Proof: U = W σ can be regarded as an element in L(H ⊗ K) (not necessary a unitary) such that V12 U13 U23 = U23 V12 . Suppose that SU is the subalgebra {(ω ⊗ id)(U ) : ω ∈ L(H)∗ } of L(K). Then a similar argument as in [2, A.3(d) or 3.6(d)] shows that {(s ⊗ 1)U (1 ⊗ a) : s ∈ SˆV ; a ∈ SU } ⊆ SˆV ⊗ SU ⊆ M (SˆV ⊗ K(K)). Hence if K 0 = SU (K) is dense in K, then W is non-degenerate in the above sense (note that (1 ⊗ s)(W (aη ⊗ ei )) = ((1 ⊗ s)W (a ⊗ 1))(η ⊗ ei ) and for any b ∈ SU ⊗ SˆV , there exist b0 ∈ SU ⊗ SˆV and r ∈ SˆV such that b = b0 (1 ⊗ r)). Next, we shall show that K 0 is dense in K if πW is non-degenerate. Indeed, since L(H)∗ separates points of M (SˆV ), L(H)∗ is σ(SˆV∗ , M (SˆV ))dense in SˆV∗ and so SU is weakly dense in πW (SˆV∗ ) (note that W ∈ M (K(K) ⊗ SˆV )). If πW is nondegenerate (i.e. πW (SˆV∗ )(K) is dense in K), then for any η ∈ K \ (0), there exist f ∈ SˆV∗ and ξ ∈ K such that ωη,ξ ((id ⊗ f )(W )) = hη, (id ⊗ f )(W )ξi = 6 0. Therefore, there exists ν ∈ L(K)∗ such that hη, (id ⊗ ν)(W )ξi = 6 0 and SU (K) is dense in K. Thus, if πW is non-degenerate, W is a non-degenerate corepresentation. Proposition and Example 2.7: Suppose that W ∈ M (K(K)⊗S) is a non-degenerate corepresentation of S. Then W defines naturally a coaction of S on Kc . Proof: Since Kc is a subspace of L(C ⊕ K), Kc ⊗ S can be regarded as a subspace of L((C ⊕ K) ⊗ S) = ¯ = 0 ⊕ η¯ · r (r ∈ S; ζ, ¯ η¯ ∈ K ⊗ S). Now for L(S ⊕ (K ⊗ S)) under the map ψ given by ψ(¯ η )(r ⊕ ζ) any x ∈ L(C ⊕ K), define δW (x) = (1S ⊕ W )(x ⊗ 1S ) ∈ L(S ⊕ (K ⊗ S)) = L((C ⊕ K) ⊗ S). δW is ¯ = 0 ⊕ W (ξ ⊗ r) (here we clearly a completely bounded map. Moreover, for any ξ ∈ Kc , δW (ξ)(r ⊕ ζ) identity ξ with its image in L(C ⊕ K)). Note that 1 ⊗ s ∈ L((C ⊕ K) ⊗ S) corresponds to the map ¯ = sr ⊕ (1 ⊗ s)ζ. ¯ Suppose that K 0 is the dense subspace of ms ∈ L(S ⊕ (K ⊗ S)) given by ms (r ⊕ ζ) K and {ei } is the approximate unit of S given by the non-degeneracy of W . It is clear that for any ˜ S (Kc ⊗ S)). η ∈ K 0 , δW (η)ms = ψ(W (η ⊗ s)) and ms δW (η) = ψ(lim(1 ⊗ s)W (η ⊗ ei )) and so δW (η) ∈ M As δW is complete bounded (and in particular bounded) and K 0 is dense in K, the restriction W of ˜ S (Kc ⊗ S) such that W (ξ) · s = W (ξ ⊗ s). It remains δW is a completely bounded map from Kc to M to show the coaction identity. In the following,P we shall not distinguish M (K(K) ⊗ S) and L(K ⊗ S). For any r, s, t, t0 ∈ S, there exists a sequence { i si ⊗ ti } in S S which converges to δ(s)(r ⊗ t). If ξ ∈ Kc , then under the isometry of Corollary 1.11(c), we have, by Lemma 1.12, the following equality in MS⊗S (Kc ⊗ S ⊗ S): X (W ⊗ id)W (ξ) · δ(s)(r ⊗ tt0 ) = lim (W ⊗ id)(W (ξ) · ti ) · (si ⊗ t0 ) i
8
=
lim
X
lim
X
(W ⊗ id)(W (ξ ⊗ ti )) · (si ⊗ t0 )
i
=
W12 (W13 (ξ ⊗ si ⊗ ti t0 )).
i
On the other hand, by Lemma 1.13(a), (id ⊗ δ)(W (ξ)) · δ(s)(r ⊗ tt0 ) = (id ⊗ δ)(W (ξ) · s) · (r ⊗ tt0 ) = (id ⊗ δ)(W (ξ ⊗ s)) · (r ⊗ tt0 ). Suppose that ξ = kη and s = s0 s00 (k ∈ K(K), η ∈ K and s0 , s00 ∈ S). Then as (idK ⊗ δ)(α(η0 ⊗ r0 )) · (s0 ⊗ t0 ) = (idK(K) ⊗ δ)(α)(η0 ⊗ δ(r0 )(s0 ⊗ t0 )) for any α ∈ K ⊗ S, η0 ∈ K and r0 , s0 , t0 ∈ S, (id ⊗ δ)(W (ξ ⊗ s)) · (r ⊗ tt0 )
=
(id ⊗ δ)(W (k ⊗ s0 )(η ⊗ s00 )) · (r ⊗ tt0 )
=
W12 W13 (k ⊗ δ(s0 ))(η ⊗ δ(s00 )(r ⊗ tt0 ))
=
W12 W13 (ξ ⊗ δ(s)(r ⊗ tt0 )).
Finally, since δ(S)(S ⊗ S) = S ⊗ S, the coaction identity is established. In the case of Hopf von Neumann algebras, we have a one to one correspondence between coactions on column Hilbert spaces and corepresentations. Let us first define what we mean by coactions and corepresentations of Hopf von Neumann algebras. From now on, until the end of this section, (S, δ) is a Hopf von Neumann algebra with predual S∗ . We shall use the usual multiplication on S∗ defined by ω · ν = (ω ⊗ ν) ◦ δ. Definition 2.8: (a) A completely bounded map from an operator space X to CB(S∗ ; X) is said to be a coaction if ((x)(ω))(ν) = (x)(ν · ω) for any x ∈ X and ω, ν ∈ S∗ . (b) If H is a Hilbert space, then W ∈ L(H)⊗S is said to be a corepresentation of S on H if (id⊗δ)(W ) = W12 W13 . Remark 2.9: (a) The above definition of coaction look a bit strange. However, if we consider X to be a weakly closed subspace of some L(H), then coactions of S on X can be translated into the following: is a completely bounded map from X to X ⊗F S such that ( ⊗ id) = (id ⊗ δ) (where ⊗F is the Fubini product which is a generalisation of the von Neumann tensor product ⊗ for von Neumann algebras). Note that X ⊗F S is completely isometrically isomorphic to CB(S∗ ; X ). (b) Suppose that S comes from a Kac algebra K and is any completely contractive coaction of S on any operator space X. Let N and U be the unit balls of X and S∗ respectively. Since ∈ CB(X; CB(S∗ ; X)) ∼ = CB(S∗ ; CB(X; X)) is a complete contraction, k(ω)(x)k ≤ 1 for any ω ∈ U and x ∈ N . It is not hard to see that this defines an action of K on N in the sense of [11, 2.2]. Example 2.10: Suppose that is a coaction of S on a von Neumann algebra M. Let N be any subset of M and XN be the closed linear span of the set {(id ⊗ ω)((x)) : ω ∈ S∗ ; x ∈ N }. We denote by N the composition of the restriction of on XN with the complete isometry from M⊗S to CB(S∗ ; M). Now we have N [(id ⊗ ω)((x))](ν) = (id ⊗ (ν · ω))((x)) ∈ XN (for any ω, ν ∈ S∗ and x ∈ N ) and it is not hard to see that N is a coaction on XN . In the case when N = x0 (x0 ∈ M), we obtain a subcomodule generated by x0 . Proposition and Example 2.11: Let H be a Hilbert space. Then there are one to one correspondences amongst the followings: (i) corepresentations of S on H; (ii) completely bounded representations of S∗ on H; (iii) completely bounded left S∗ -module structures on Hc ; (iv) coactions of S on Hc . Proof: The correspondences are actually given by the following series of identifications: L(H)⊗S ∼ = ˆ c ; Hc ) ∼ ˆ ∗ ; C) ∼ CB(L(H)∗ ⊗S = CB(Hc ; CB(S∗ ; Hc )). = CB(S∗ ; L(H)) = CB(S∗ ; CB(Hc ; Hc )) ∼ = CB(S ⊗H 9
Therefore, we need only to check the corresponding algebraic structure in each case. In particular, if we want to show that (i) induces (ii), we can easily check that for any corepresentation X, the induced map ρX in CB(S∗ ; L(H)) defined by ρX (ω) = (id ⊗ ω)(X) is a representation. For (ii) induces (iii): ˆ c ; Hc ) if ρ is a completely bounded representation of S∗ , then the corresponding map mρ in CB(S ⊗H defined by mρ (ω ⊗ξ) = ρ(ω)ξ is obviously a completely bounded left S∗ -module structure. Now suppose that m is a completely bounded left S∗ -module structure on Hc . Then it is not hard to see that the corresponding element m in CB(Hc ; CB(S∗ ; Hc )) is given by m (ξ)ω = m(ω ⊗ξ) and is a coaction. This shows that (iii) induces (iv). Finally, given a coaction of S on Hc , we would like to check that the corresponding element W in L(H)⊗S is a corepresentation. In fact, we have (ξ)(ω) = (id ⊗ ω)(W )ξ. Hence (id ⊗ (ω ⊗ ν) ◦ δ)(W )ξ = (id ⊗ ω)(W )((id ⊗ ν)(W )ξ) = (id ⊗ ω ⊗ ν)((W )12 (W )13 )ξ. This completes the proof. Note that (iii) and (iv) are equivalent for any operator space X (instead of Hc ).
III. Crossed products of operator algebras / operator spaces Throughout this section, unless specified, (H, V, U ) is a Kac system (see [2]). In fact, most of the results in this section holds for a more general situation of l-standard (or r-standard) representations (see [16, 4.9(a)]) except the equivariant property of the map Ψ in Theorem 3.8 which requires Kac representations (see [16, 4.9(b)]). However, for the ease of the presentation, we shall not recall these here. For simplicity, we denote S = SV , Sˆ = SˆV , L = LV , ρ = ρV , λ = Ad(U )ρV and R = Ad(U )LV (see [2]). As in [2, 6.1], we let V˜ = (R ⊗ id)V σ and Vˆ = (id ⊗ λ)V σ (where σ is the flip of the two variables). Moreover, we denote by C the space of compact operator on H (note that K means K(l2 ) in this paper). Then idX ⊗ L will map MS (X ⊗ S) to MC (X ⊗ C) (by Lemma 1.13(a)) and MC (X ⊗ C) is an operator L(H)-bimodule (Corollary 1.11(b)). Note L and ρ are forgettable embeddings of S and Sˆ respectively which may sometimes be omitted. Definition 3.1: (a) Suppose that is a coaction of S on X. Then the closed linear span of the subset ˆ of MC (X ⊗ C) is called the reduced crossed product of and is {(idX ⊗ L)(x) · ρ(t) : x ∈ X; t ∈ S} ˆ denoted by X ×,r S. (b) Suppose that 0 is a coaction of Sˆ on X. Then the closed linear span of the subset {(idX ⊗ ρ)0 (x) · R(s) : x ∈ X; s ∈ S} of MC (X ⊗C) is called the reduced crossed product of 0 and is denoted by X ×0 ,r S. Note that our definition for reduced crossed product of coaction of Sˆ is slightly different from that of [2, 7.1]. However, Ad(U ) will define a complete isometry between these two definitions of crossed products (by Lemma 1.13(a)). The benefit of defining it as above is that some of the notations and proofs become simpler. Before we formulate and prove the main theorem of this section, we shall first recall from [2] the following facts about Kac-system. ˆ Lemma 3.2: Let s ∈ S and t ∈ S. ˆ (regard V˜ ∈ M (C ⊗ S)). ˆ (a) V˜ commutes with L(s) ⊗ 1 and Ad(V˜ )(ρ(t) ⊗ 1) = (ρ ⊗ id)δ(t) ∗ ˆ (b) V commutes with 1 ⊗ R(s), Ad(V )(L(x) ⊗ 1) = (L ⊗ L)δ(s) and Ad(V )(1 ⊗ ρ(t)) = (ρ ⊗ ρ)δ(t). ˆ ˆ ˆ (c) V commutes with 1 ⊗ ρ(t), Ad(V )((id ⊗ L)δ(s)) = 1 ⊗ L(s) and Ad(V )(1 ⊗ R(s)) = (id ⊗ R)(δ(s)σ ) (regard Vˆ ∈ M (S ⊗ C)) where σ is again the flip of the two variables. ˆ s ∈ S} is dense in C. (d) The linear span of {ρ(t)R(s) : t ∈ S; In fact, part (a) comes from [2, 6.8(b)&(c)] (note that Ad(U ⊗ U )(Vˆ ) = V˜ ) and part (b) is clear. The final equality of part (c) follows from the first equality of part (b). Part (d) is a reformulation of [2, 6.10]. 10
ˆ Lemma 3.3: X×,r Sˆ is an essential operator S-bimodule. Moreover, there exists a canonical completely ˆ bounded map jX from X to MSˆ (X ×,r S) such that jX (x) · t = (idX ⊗ L)(x) · ρ(t) and t · jX (x) = ˆ The same is true for X ×0 ,r S. ρ(t) · (idX ⊗ L)(x) (x ∈ X; t ∈ S). Proof: Let X be a subspace of L(K). Then MC (X ⊗C) is naturally a subspace of L(K ⊗H) and 1⊗ρ is a non-degenerate representation of Sˆ on L(K ⊗ H). Moreover, X ×,r Sˆ is generated by elements of the ˆ form (id ⊗ L)(x)(1 ⊗ ρ(t)) and hence is an essential operator right S-module. On the other hand, for any ω ∈ L(H)∗ , we have (id ⊗ id ⊗ ω)(V23 ((id ⊗ L)(x) ⊗ 1)) = (id ⊗ id ⊗ ω)((id ⊗ L ⊗ L)( ⊗ id)(x)V23 ). Hence (1⊗ρ(t))(id⊗L)(x) can be approximated by sums of elements of the form (y)(1⊗ρ(r)) (because ω = ν · L(s) for some s ∈ S and ν ∈ L(H)∗ and (1 ⊗ s)(x) ∈ X ⊗ S). This shows that X ×,r Sˆ is an ˆ essential S-bimodule. Now let jX be the map (idX ⊗ L) (from X to MC (X ⊗ C))), then Proposition ˆ and satisfies the required conditions. 1.9 implies that jX (x) ∈ MSˆ (X ×,r S) ˆ for any x Let ¯ be a map defined by ¯(¯ x) = V˜ · (¯ x ⊗ 1) · V˜ ∗ ∈ MC⊗Sˆ (X ⊗ C ⊗ S) ¯ ∈ MC (X ⊗ C) (see ˆ Remark 1.14(b)). Then by Lemma 3.2(a), ¯ induces a dual coaction on X ×,r S in the following sense (note that the corresponding dual coaction on X ×0 ,r S is given by Ad(Vˆ σ ) and Lemma 3.2(c)). Lemma 3.4: There exists a dual coaction ˆ of Sˆ on X ×,r Sˆ such that ˆ((idX ⊗ L)(x) · ρ(t)) = ˆ (x ∈ X; t ∈ S). ˆ The same is true for coaction of Sˆ (in which (idX ⊗ L ⊗ idSˆ )((x) ⊗ 1) · (ρ ⊗ id)δ(t) 0 0 0 case ˆ ((idX ⊗ ρ) (x) · R(s)) = ((idX ⊗ ρ) (x) ⊗ 1) · (R ⊗ id)δ(s)). Moreover, ˆ is always a completely isometric (2-sided) non-degenerate coaction. Remark 3.5 Note that if V is only a regular multiplicative unitary instead of coming from a Kac system and if we consider only the crossed products of coactions by S, then we still have Lemmas 3.3 ˆ Moreover, the last two equalities of Lemma 3.2(b) still hold in this case. and 3.4 for X ×,r S. Lemma 3.6: Suppose that X and Y are coactions on X and Y respectively by S. If T is a completely bounded equivariant map from X to Y , then T induces a completely bounded equivariant map T × id from X ×X ,r Sˆ to Y ×Y ,r Sˆ such that (T × id)((idX ⊗ L)X (x) · ρ(t)) = (idX ⊗ L)Y (T (x)) · ρ(t). Proof: By Lemma 1.12, T induces a completely bounded map T ⊗ id from MC (X ⊗ C) to MC (Y ⊗ C). It is clear that the restriction of it on X × ,r Sˆ is the required map (as T is equivariant). X
Example 3.7: Let E be a Hilbert A-module with coaction δE by S. Suppose that Ec is the column operator space structure on E and E be the coaction on Ec induced by δE (see Lemma 2.4). Then ˆ c (where E ×δ ,r Sˆ is as defined in [1, 6.6]). Moreover, the dual coactions in Ec ×E ,r Sˆ = (E ×δE ,r S) E both cases also match with each other. Similar to [2, §7], we can define a completely bounded map ˇ on X ⊗ C by ˇ(x ⊗ k) = Vˆ σ · ((x)13 · (k ⊗ 1)) · Vˆ σ∗ ∈ MC⊗S (X ⊗ C ⊗ S) for any x ∈ X and k ∈ C. We also have the following analogue of [2, 7.5]. The idea of the proof is similar to the case of C ∗ -algebras. Theorem 3.8: Let be a right non-degenerate coaction of S on X and ˜ be the bidual coaction on the ˆ ׈,r S. Then there exists a completely bounded map Ψ from X ⊗ C double crossed product (X ×,r S) ˆ to (X ×,r S) ׈,r S that respects ˜ and ˇ. Moreover, if is a complete isometry, then Ψ is a completely ˆ ׈,r S. isometric isomorphism from X ⊗ C onto (X ×,r S) ˆ s ∈ S) generate C. As Proof: First of all, by Lemma 3.2(d), elements of the form ρ(t)R(s) (t ∈ S; is a right non-degenerate coaction and L is a non-degenerate representation, elements of the form ˆ generate X ⊗ C. Let Φ be the complete isometry from (id ⊗ L)(x) · ρ(t)R(s) (x ∈ X; s ∈ S; t ∈ S) MC⊗C (X ⊗ C ⊗ C) to itself induced by Ad(V ∗ ) (see Lemma 1.13(a)). Then the completely bounded map Ψ = Φ ◦ ((id ⊗ L) ⊗ id) from MC (X ⊗ C) to MC⊗C (X ⊗ C ⊗ C) will send (id ⊗ L)(x) · ρ(t)R(s) to ˆ ((id ⊗ L)(x) ⊗ 1) · ((ρ ⊗ ρ)δ(t)(1 ⊗ R(s))) (by Lemma 3.2(b)). It is clear that elements of the this form ˆ ׈,r S and Ψ(X ⊗ C) is dense in (X ×,r S) ˆ ׈,r S. It remains to show that Ψ is generate (X ×,r S) ˆ equivariant. Note that ˜ maps ((id ⊗ L)(x) ⊗ 1) · ((ρ ⊗ ρ)δ(t)(1 ⊗ R(s))) to ((id ⊗ L)(x) ⊗ 1 ⊗ 1) · ((ρ ⊗ 11
ˆ ⊗ 1)(1 ⊗ (R ⊗ id)δ(s)). On the other hand, by Lemma 3.2(c), ˇ sends (id ⊗ L)(x) · ρ(t)R(s) to ρ)δ(t) ((id ⊗ L)(x) ⊗ 1) · (ρ(t) ⊗ 1)(R ⊗ id)δ(s). Now Ψ ⊗ id will clearly send this element to the former one. ˆ ׈,r S. Finally, if is a complete isometry, then so is Ψ and Ψ(X ⊗ C) = (X ×,r S) In the case of operator algebras, we have the following result. Theorem 3.9: Let B be an operator algebra with coaction by S. ˆ (a) B ×,r Sˆ is an operator algebra as well as an essential operator S-bimodule and there exists a ˆ satisfying the conditions in canonical completely bounded homomorphism jB from B to MSˆ (B ×,r S) Lemma 3.3. (b) If B 0 is another operator algebra with coaction γ by S and T is a completely bounded equivariant homomorphism from B to B 0 , then the map T × id from B ×,r Sˆ to B 0 ×γ,r Sˆ given by Lemma 3.6 is an algebraic homomorphism. (c) The dual coaction ˆ on B ×,r Sˆ (see Lemma 3.4) is an algebraic homomorphism. Moreover, if is right non-degenerate, then there is a completely bounded equivariant homomorphism Ψ from B ⊗ C ˆ ׈,r S and Ψ is a completely isometric algebraic isomorphism whenever is completely to (B ×,r S) isometric. Remark 3.10: (a) Note that even in the case of completely bounded actions of locally compact groups on operator algebras, this duality result is seemingly new. (b) If T is an equivariant completely bounded map from X to Y , then ΨY ◦(T ⊗idC ) = ((T ×id)×id)◦ΨX where ΨX and ΨY are the maps given by Theorem 3.8. ˆ By Lemmas 3.3 and 3.4, the reduced crossed product is a right Hopf S-bimodule in the following ∗ 0 0 sense: for any Hopf C -algebra S , an essential operator S -bimodule X is said to be a right Hopf S 0 bimodule if there exists a coaction of S 0 on X such that (x · t) = (x) · δ(t) and (t · x) = δ(t) · (x) for any x ∈ X and t ∈ S 0 (note that this terminology follows from that of Hopf algebras). It is interesting to ˆ note that any representation or non-degenerate corepresentation of S induces a right Hopf S-bimodule. In fact, any non-degenerate corepresentation of S will induce a coaction (see Example 2.7) and hence a crossed product. On the other hand, any representation of S will induces a unitary corepresentation ˆ The corresponding double crossed product will be a right Hopf of Sˆ and hence a coaction (of S). ˆ S-bimodule. ˆ We end this section with the following natural question: does every right Hopf S-bimodule come from a reduced crossed product? Note that this question is a general form of the Landstad duality for crossed products of actions of locally compact groups (see [12]).
IV. Crossed products and operator exactness of operator spaces Throughout this section, i and q are completely bounded maps from X to Y and from Y to Z respectively. The sequence: q i 0→X→Y →Z→0 (3) is said to be operator exact if i is a complete topological injection and q is a complete topological surjection such that Ker(q) = i(X). Note that it is difference from the notion of exact sequence of operator space in [20]. Definition 4.1: (a) An operator space E is said to be operator exact if the sequence 0 → K ⊗ E → B ⊗ E → (B/K) ⊗ E → 0 is operator exact.
12
(b) An operator algebra B is said to be operator exact if the underlying operator space is operator exact. It is clear that if B is a C ∗ -algebra, then B is an operator exact if and only if it is C ∗ -exact. Moreover, if E is an operator exact operator space, then it is exact in the sense of [20] (see Remark 4.7 for a more detail comparison). Lemma 4.2: If A is a C ∗ -algebra such that the sequence q⊗id
i⊗id
0 → X ⊗ A −→ Y ⊗ A −→ Z ⊗ A → 0 is operator exact, then so is (3). Proof: It is clear that i ◦ q = 0. Take any a ∈ A+ and f ∈ A∗+ such that kak ≤ 1 and f (a) = 1. Since q ⊗ idA is a complete topological injection, there exists µ > 0 such that UZ⊗A⊗K ⊆ µ(q ⊗ idA ⊗ idK )(UY ⊗A⊗K ). Hence for any z¯ ∈ Z ⊗ K with k¯ z k ≤ 1, k¯ z ⊗ ak ≤ 1 and there exists y˜ ∈ Y ⊗ K ⊗ A with k˜ y k ≤ µ such that (q ⊗ idK ⊗ idA )(˜ y ) = z¯ ⊗ a. This shows that UZ⊗K ⊆ kf kµUY ⊗K and q is a complete topological surjection. On the other hand, since i ⊗ idA is a complete topological injection, there exists λ > 0 such that for any x ¯ ∈ X ⊗ K, k¯ xk = k¯ x ⊗ ak ≥ λk(i ⊗ idK ⊗ idA )(¯ x ⊗ a)k = λk(i ⊗ idK )(¯ x)k and i is a complete topological injection. Finally, it is not hard to see that Ker(q) ⊆ Im(i). Before proving the main theorem of this section, we want to recall from Remark 3.5 that we have some nice properties for the crossed products of coactions by S even in this case when V is only a regular multiplicative unitary. Theorem 4.3: Let V be a regular multiplicative unitary and X , Y and Z be coactions of S = SV on X, Y and Z respectively. Suppose that i and q are equivariant. (a) If V is amenable (see [2] or [15]) and the sequence (3) is operator exact, then so is the sequence q×id i×id 0 → X ×X ,r Sˆ −→ Y ×Y ,r Sˆ −→ Z ×Z ,r Sˆ → 0.
(4)
(b) If V is irreducible and coamenable and the sequence (4) is operator exact, then so is (3). Proof: (a) Let X 0 = i(X) and X 0 be the coaction on X 0 induced by X . It is clear that X 0 is the “restriction” of Y . Suppose that P = Y ×Y ,r Sˆ and Q = X 0 ×X 0 ,r Sˆ (⊆ P ) and consider the complete isomorphism from MC (X ⊗ C) to MC (X 0 ⊗ C) ⊆ MC (Y ⊗ C) (where C = K(H)). Then i × id is the ˆ Moreover, since (q × id) ◦ (i × id) = 0, q × id restriction of this complete isomorphism on X ×X ,r S. induces a completely bounded map ϕ from P/Q to Z ×Z ,r Sˆ (see Remark 1.5(c)). It remains to show that ϕ is a complete isomorphism. Let jY : Y 7→ MSˆ (P ) be the completely bounded map given by ˆ Lemma 3.3 and qP be the completely bounded S-bilinear map from MSˆ (P ) to MSˆ (P/Q) (by Lemma 0 0 ˆ 1.15). It is clear that X ⊆ Ker(qP ◦ jY ) (as qP (jY (X )) · S = (0)). By Remark 1.5(c) and the operator exactness of (3), we obtain a completely bounded map π from Z to MSˆ (P/Q) such that π(q(y)) = qP (jY (y)). The restriction of π ⊗ idC induces a completely bounded map π ˜ from Z ×Z ,r Sˆ to MS⊗C ((P/Q) ⊗ C) ˆ such that π ˜ (jZ (z) · t) = (π ⊗ idC )((idZ ⊗ L)Z (z)) · (1 ⊗ ρ(t)) (see Lemma 1.12 and Corollary 1.11(c)). On the other hand, let ˜jY and q˜P be the maps from MC (Y ⊗ C) to MS⊗C (P ⊗ C) (again by Lemma ˆ 1.12 and Corollary 1.11(c)) and from MS⊗C (P ⊗ C) to M ((P/Q) ⊗ C) respectively. It is clear that ˆ ˆ S⊗C ˜ (π ⊗ idC ) ◦ (q ⊗ idC ) = q˜P ◦ jY on MC (Y ⊗ C) (by Lemma 1.13(b)) and so π ˜ ((idZ ⊗ L)Z (q(y)) · ρ(t)) = q˜P (˜jY ((idY ⊗ L)Y (y)) · (1 ⊗ ρ(t))) (as q is equivariant). Now if Y is a closed subspace of L(K), MS⊗C (P ⊗ C) can be identified as a ˆ ˆ subspace of L(K ⊗ H ⊗ H). If we regard V as an element of M (S ⊗ C), then, as q˜P is M (Sˆ ⊗ C)-bilinear, 13
we have, by Lemma 3.2(b) (see also Remark 3.5), Ad(V ∗ )[˜ π ((idZ ⊗ L)Z (q(y)) · ρ(t))] ˆ = q˜P [(1 ⊗ V ∗ )(idK(K) ⊗ L ⊗ L)(idK(K) ⊗ δ)Y (y)(1 ⊗ V )(1 ⊗ (ρ ⊗ ρ)δ(t))] ˆ = q˜P [((idK(K) ⊗ L)Y (y) ⊗ 1)(1 ⊗ (ρ ⊗ ρ)δ(t))] =
ˆ q˜P [(jY (y) ⊗ 1) · (idSˆ ⊗ ρ)δ(t)].
(5)
ˆ to MC ((P/Q) ⊗ C) ⊆ Let ψ1 and ψ2 be the forgettable complete isometries from MSˆ ((P/Q) ⊗ S) ˆ MS⊗C ((P/Q) ⊗ C) (see Corollary 1.11(c)) and from MSˆ (P ⊗ S) to MS⊗C (P ⊗ C) respectively. Then ˆ ˆ ˆ ∈ M ˆ (P ⊗ S) ˆ (considered as a subspace of M ˆ (Y ⊗ C ⊗ S)) ˆ and its ((idY ⊗ L)Y (y) ⊗ 1) · (ρ ⊗ idSˆ )δ(t) S C⊗S ˆ image under ψ2 is (jY (y) ⊗ 1) · (idSˆ ⊗ ρ)δ(t). Moreover, we have q˜P ◦ ψ2 = ψ1 ◦ (qP ⊗ idSˆ ) (by Lemma 1.13(b)) and from (5), ˆ ⊆ q˜P (ψ2 (M ˆ (P ⊗ S))) ˆ ⊆ ψ1 ((qP ⊗ id ˆ )(M ˆ (P ⊗ S))). ˆ Ad(V ∗ )(˜ π (Z ×Z ,r S)) S S S Since V is amenable, Sˆ has a counit u which induces completely bounded maps uP and uP/Q from ˆ to P and from M ˆ ((P/Q) ⊗ S) ˆ to P/Q respectively (using Lemma 1.13(a)). Note that MSˆ (P ⊗ S) S uP/Q ◦ (qP ⊗ idSˆ ) = qP ◦ uP (by Remark 1.14(a)) and uP is the restriction of (idK(K) ⊗ idC ⊗ u) on ˆ ˆ (considered as a subspace of M (K(K)⊗C⊗S)). ˆ Thus, uP (((idY ⊗L)Y (y)⊗1)·(ρ⊗id ˆ )δ(t)) MSˆ (P ⊗S) = S −1 ∗ (idY ⊗ L)Y (y) · ρ(t). Hence if j is the completely bounded map uP/Q ◦ ψ1 ◦ Ad(V ) ◦ π ˜ from Z ×Z ,r Sˆ to P/Q, then j((id ⊗ L)Z (q(y)) · ρ(t))
ˆ = uP/Q ◦ ψ1−1 ◦ q˜P [(jY (y) ⊗ 1) · (idSˆ ⊗ ρ)δ(t)] ˆ uP/Q ◦ ψ1−1 ◦ q˜P ◦ ψ2 (((idY ⊗ L)Y (y) ⊗ 1) · (ρ ⊗ idSˆ )δ(t)) ˆ = uP/Q ◦ (qP ⊗ idSˆ )(((idY ⊗ L)Y (y) ⊗ 1) · (ρ ⊗ idSˆ )δ(t)) ˆ = qP ◦ uP (((idY ⊗ L)Y (y) ⊗ 1) · (ρ ⊗ id ˆ )δ(t))
=
S
= qP ((idY ⊗ L)Y (y) · ρ(t)) and j is an inverse of ϕ. (b) Suppose that the sequence (4) is operator exactness. By part (a) as well as Theorem 3.8 and Remark 3.10(b), we have the following operator exact sequence: q⊗id
i⊗id
0 → X ⊗ C −→ Y ⊗ C −→ Y /X ⊗ C → 0. Now Lemma 4.2 shows that (3) is operator exact. If G is an amenable locally compact group with completely bounded actions αX , αY and αZ on X, Y and Z respectively (see Example 2.3(b)) such that i and q are equivariant, then the sequence (3) is operator exact if and only if the sequence ˆ → Y ×α ,r G ˆ → Z ×α ,r G ˆ→0 0 → X ×αX ,r G Y Z is operator exact. Note that part (a) of the above Theorem actually holds for any amenable X-covariant representation (where (T, X, S) is a Fourier duality or equivalently, S is a coamenable Hopf C ∗ -algebra; see [17]). This applies, in particular, to the case of amenable manageable multiplicative unitaries (or more generally, amenable C ∗ -multiplicative unitaries; see [16]). We can now use exactly the same sort of argument as in the case of C ∗ -algebras to obtain the following theorem. 14
Theorem 4.4: Let B be an operator exact operator algebra and V be a regular amenable multiplicative unitary. For any coaction B of S = SV on B, B ×B ,r Sˆ is also operator exact. If, in addition, V is irreducible and coamenable, then B is operator exact if and only if B ×B ,r Sˆ is operator exact. Proof: For any operator space X, B induces a coaction idX ⊗B on X⊗B such that (X ⊗ B) ×idX ⊗B ,r Sˆ ˆ (as subspaces of MC (X ⊗B⊗C)). Hence by Theorem 4.3(a) and the operator exactness = X ⊗(B×B ,r S) of B, we have the operator exact sequence: 0 → (K ⊗ B) ×idK ⊗B ,r Sˆ → (B ⊗ B) ×idB ⊗B ,r Sˆ → (B/K ⊗ B) ×idB /K⊗B ,r Sˆ → 0. It is easily seen that this gives us the required operator exact sequence: ˆ → B ⊗ (B × ,r S) ˆ → B/K ⊗ (B × ,r S) ˆ →0 0 → K ⊗ (B ×B ,r S) B B and B ×B ,r Sˆ is operator exact. The finally statement follows from Theorem 4.3(b). Remark 4.5: In fact, the same result holds for operator spaces (using exactly the same proof) but we formulate it in term of operator algebras to make it look more comfortable. In [20], Pisier define the “degree of exactness” for exact operator space. Similarly, we can define a stable degree of exactness for any operator exact operator space E as follows: ex(E) := kT −1 kcb (where T is the map from (B ⊗ E)/(K ⊗ E) to (B/K) ⊗ E as given by Definition 4.1). If E is an operator exact operator space, then E is exact and ex(E) ≤ ex(E). It is natural to ask whether there is any relation between the stable degree of exactness of the crossed product and that of the original operator space. Proposition 4.6: Let V be a regular amenable multiplicative unitary. If is a coaction of SV on E, then ex(E ×,r SˆV ) ≤ ex(E)kkcb (see Remark 4.5). Proof: Let F = E ×,r SˆV and T be the canonical map from (B ⊗ F )/(K ⊗ F ) to (B/K) ⊗ F . By the proof of Theorem 4.4 (see Remark 4.5), T is the canonical map from ((B ⊗ E) ×id⊗,r SˆV )/((K ⊗ E) ×id⊗,r SˆV ) to ((B/K) ⊗ E) ×id⊗,r SˆV . Now in the proof of Theorem 4.3(a), the inverse of this map is of the form u(B⊗F )/(K⊗F ) ◦ ψ1−1 ◦ Ad(V ∗ ) ◦ π ˜ . Hence kT −1 kcb ≤ k˜ π kcb (note that ψ1 is a complete isometry). Recall that π ˜ is the restriction of π ⊗ id and π is the composition of the canonical map R from (B/K)⊗E to (B ⊗E)/(K⊗E) (so kRkcb = ex(E)) and the canonical map R0 from (B ⊗E)/(K⊗E) to MSˆ ((B ⊗ F )/(K ⊗ F )). It remains to show that kR0 kcb ≤ kkcb . Let J be the map from B ⊗ E to MSˆ ((B ⊗ E) ×id⊗,r SˆV ) given by Lemma 3.3 (hence kJkcb ≤ k(idB⊗E ⊗ L) ◦ (idB ⊗ )kcb ≤ kkcb ). By Lemma 1.6 and the proof of Lemma 1.12, the map Q from MSˆ (B ⊗ F ) to MSˆ ((B ⊗ F )/(K ⊗ F )) is a complete contraction. Since R0 is induced by Q◦J (K ⊗E ⊆ Ker(Q◦J)), kR0 kcb ≤ kQkcb kJkcb ≤ kkcb (see Remark 1.5(c)). We end this paper with the following comparison between exactness and operator exactness. Remark 4.7: Note that a sequence q
i
0→X→Y →Z→0 is operator exact if and only if the sequence: q⊗id
i⊗id
0 → X ⊗ K −→ Y ⊗ K −→ Z ⊗ K → 0 is exact in the sense of [20]. Therefore, an operator space E is operator exact if and only if E ⊗ K is exact (in the sense of [20]). In this case, ex(E) = ex(E ⊗ K). Consequently, if E is stable in the sense that E ⊗ K is completely isomorphic to E, then exactness and operator exactness of E coincide.
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Reference: [1] S. Baaj and G. Skandalis, C ∗ -alg`ebres de Hopf et th´eorie de Kasparov ´equivariante, K-theory 2(1989), 638-721. [2] S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualit´e pour les produits crois´es de C ∗ -alg`ebres, ´ Norm. Sup., 4e s´erie, t. 26 (1993), 425-488. Ann. scient. Ec. [3] D.P. Blecher, The standard dual of an operator space, Pacific J. Math. 153(1992), 15-30. [4] D. P. Blecher, A new approach to Hilbert C ∗ -modules, Math. Ann. 307(1997), 253-290. [5] D.P. Blecher and V.I. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99(1991), 262-292. [6] D.P. Blecher, Z.-J. Ruan and A.M. Sinclair, A characterization of Operator algebras, J. Funct. Anal. 89 (1990), 188-201. [7] E. Christensen and A. Sinclair, A survey of completely bounded Operators, Bull. London Math. Soc. 21(1989), 417-448. [8] E. Christensen, E.G. Effros and A. Sinclair, Completely bounded multilinear maps and C ∗ -algebraic cohomologoy, Invent. Math. 90(1987), 279-296. [9] E. Effros and Z. J. Ruan, On approximation properties for operator spaces, International J. Math. 1(1990), 163-187. [10] E. Effros and Z. J. Ruan, A new approach to Operator Spaces, Canad. Math. Bull. 34(1991), 329-337. [11] M. Enock and J.-M. Schwartz, Algebres de Kac Moyennables, Pacific Journal of Mathematics 125(1986), 363-379. [12] M.B. Landstad, Duality theory for covariant systems, Trans. Amer. Math. Soc. 248(1979), 223-267. [13] P.S. Muhly, A finite-dimensional introduction to Operator algebras, Operation algebras and Applications (Samos, 1996), SE: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 495, 313-354. [14] C.K. Ng, Discrete coaction on C ∗ -algebras, J. Austral. Math. Soc. (Series A) 60(1996), 118-127. [15] C.K. Ng, Coactions and crossed products of Hopf C ∗ -algebras, Proc. London Math. Soc.(3) 72(1996), 638-656. [16] C.K. Ng, Duality of Hopf C ∗ -algebras, preprint. [17] C.K. Ng, Amenability of Hopf C ∗ -algebras, preprint. [18] T.W. Palmer, Banach algebras and the general theory of ∗-algebras Vol. I: Algebras and Banach algebras, Encyclopedia of Math. 1994, Camb. Univ. Press. [19] G. K. Pedersen, C ∗ -algebras and their automorphism groups, 1979, Academic Press. [20] G. Pisier, Exact Operator spaces, Recent advances in operator algebras (Orleans, 1992), Asterisque [Asterisque] No. 232(1995), 159–186. [21] Z. J. Ruan, Subspaces of C ∗ -algebras, J. Funct. Anal. 76(1988), 217-230. [22] Z. J. Ruan, On the predual of dual algebras, J. Oper. Theory 27(1992), 179-192. [23] A.M. Sinclair and R.R. Smith, Hochschild cohomology of von Neumann algebras, London Math. Soc. LNM 203, 1995, Cambridge University Press. Address: Mathematical Institute, 24-29 St. Giles, Oxford OX1 3LB, United Kingdom. Email address:
[email protected]
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