PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 7, July 1997, Pages 2109–2113 S 0002-9939(97)03792-1
A HILBERT C*-MODULE METHOD FOR MORITA EQUIVALENCE OF TWISTED CROSSED PRODUCTS HUU HUNG BUI (Communicated by Palle E. T. Jorgensen)
Abstract. We present a new proof for Morita equivalence of twisted crossed products by coactions within the abstract context of crossed products of Hilbert C ∗ -modules. In this context we are free from representing all C ∗ -algebras and Hilbert C ∗ -modules on Hilbert spaces.
The notion of Morita equivalence of twisted coactions was introduced in [B]. In [B, Theorem 3.3] we established conditions on twisted coactions which are sufficient to ensure Morita equivalence of the corresponding crossed product C ∗ -algebras. Later [ER] gave a shorter proof for this result using their results on multipliers of imprimitivity bimodules. However in the proofs of both [B] and [ER], all C ∗ algebras and Hilbert C ∗ -modules need to be represented on Hilbert spaces. In this paper we present a new proof for [B, Theorem 3.3] based on the notion of crossed products of Hilbert C ∗ -modules introduced in [B2]. Crossed products of Hilbert C ∗ -modules in [B2] were defined as subspaces of adjointable operators between Hilbert C ∗ -modules. In this abstract context, we are free from representing all C ∗ -algebras and Hilbert C ∗ -modules on Hilbert spaces as in [B] and [ER]. As a consequence, the proof here is shorter and more elegant than that of [B]. Our approach is close to the spirit of [BS], and different from [ER]. Throughout this paper G is a locally compact group and N is a closed normal amenable subgroup of G. Recall from [M, Lemma 3] that there is a surjective homomorphism Ψ from Cr∗ (G) into Cr∗ (G/N ) such that Ψ(λG (r)) = λG/N (qN (r)), where qN : G → G/N is the quotient map, λG and λG/N are the left regular representations of G and G/N . We denote by WG the unitary operator on L2 (G×G) defined by [WG ξ](r, s) = ξ(r, r−1 s). If f is an element of the Fourier algebra A(G), then Sf (WG ) = Mf . Here Sf denotes the slice map, see [LPRS, §1]. To apply [B2, Theorem 1.6] to this paper, we need to show that WG is a regular multiplicative unitary. For any ξ, η ∈ L2 (G), we define ∀T ∈ B(L2 (G)).
ωη,ξ = hT ξ|ηi, Then for any ω = ωη,ξ , we have
h(id ⊗ ω)(WG )ξ 0 |η 0 i = hMω◦λG ξ 0 |η 0 i,
∀ξ 0 , η 0 ∈ L2 (G).
It then follows that SbWG = C0 (G), and the crossed product of [B2, Proposition 1.5] is just the crossed product of [LPRS, Definition 2.4]. The unitary operator Received by the editors October 23, 1995 and, in revised form, February 6, 1996. 1991 Mathematics Subject Classification. Primary 46L05, 22D25. c
1997 American Mathematical Society
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HUU HUNG BUI
ξ 7−→ ξ from L2 (G) onto the conjugate space L2 (G) satisfies the conditions of [BS2, Exemples 3.4.3], and hence WG is regular. Let (δD , WD ) be a twisted coaction of (G, G/N ) on a C ∗ -algebra D in the sense of [PR, Definition 2.1]. Put IWD =
\
{ ker(π × µ) : (π, µ) is a covariant representation of (D, G, δD ) which preserves WD }.
The twisted crossed product D ×δD ,WD G is the quotient D ×δD G/IWD ; see [PR, Definition 2.8]. Put nD (f ) = δ¯D (Sf {WD }) − [1D ⊗ Mf ◦qN ],
∀f ∈ A(G/N ).
Recall from [B, Lemma 3.5] that IWD is the closed subspace generated by γnD (f )γ 0 for all γ, γ 0 ∈ D ×δD G and f ∈ A(G/N ). For convenience we recall the following definition from [B, Definition 3.2]. Definition 1. Let (δA , WA ) and (δB , WB ) be twisted coactions of (G, G/N ) on C ∗ -algebras A and B. We say that (δA , WA ) and (δB , WB ) are Morita equivalent if there are an A, B-imprimitivity bimodule X and a δB -compatible coaction δX of G on X such that ˆ ¯) ◦ δA (Ahx|yi), ∀x, y ∈ X, (i) δX (x)δX (y)∗ = (ϑ ⊗id ∗ ¯ ˆ ˆ ) ◦ δX (x) = WA (x ⊗1)W , ∀x ∈ X, (ii) (idX ⊗Ψ B where ϑ : A → K(X) is the natural isomorphism. Next we will give a new proof for the following result [B, Theorem 3.3]. Theorem 2. If the twisted coactions (δA , WA ) and (δB , WB ) are Morita equivalent by means of (X, δX ), then the twisted crossed products A×δA ,WA G and B ×δB ,WB G are Morita equivalent. Put E = K(X) and J = K(X ⊕ B). We will use the notation δE , δJ , c¯ij and d¯ij of [B2]. Put WE = (ϑ ⊗ id¯)(WA ). Then (δE , WE ) is a twisted coaction of (G, G/N ) on E, and E ×δE ,WE G is isomorphic to A ×δA ,WA G; see [B, Lemma 3.4]. Therefore we may assume that A = E, WA = WE and δA = δE . Lemma 3. (i)
Each element nA (f )δX (x) is the limit of finite sums n X
δX (yi )nB (gi ),
yi ∈ X, gi ∈ A(G/N ).
i=1
(ii)
Each element δX (y)nB (g)∗ is the limit of finite sums n X
nA (fi )∗ δX (xi ),
xi ∈ X, fi ∈ A(G/N ).
i=1
Proof. (i) We write f = g · Ψ(u) for some g ∈ A(G/N ) and u ∈ Cr∗ (G). We shall ˆ r∗ (G), it is the limit of finite sums denote c¯ij (m) by mc . Since [1A ⊗ u]δX (x) ∈ X ⊗C
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MORITA EQUIVALENCE OF CROSSED PRODUCTS
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ˆ i . We compute yi ⊗v d¯12 δ¯A (Sf {WA })δX (x) = δ¯J (¯ c11 (Sf {WA }))δJ (xc ) = Sf {(δJ ⊗ id¯) ◦ (c11 ⊗ id¯)(WA )[δJ (xc ) ⊗ 1]} ˆ ˆ ¯)(WA [x ⊗1]} = Sf {(δJ ⊗ id¯) ◦ (c12 ⊗id ˆ ¯)((idX ⊗Ψ ˆ ¯) ◦ δX (x)WB )} = Sg {[1J ⊗ Ψ(u)](δJ ⊗ id¯) ◦ (c12 ⊗id ¯ ˆ = Sg {(δJ ⊗ Ψ) ◦ (c12 ⊗id)([1 A ⊗ u]δX (x))(δJ ◦ c22 ⊗ id)(WB )},
which is the limit of finite sums n X ˆ ˆ i )(δJ ◦ c22 ⊗ id¯)(WB )} Sg {(δJ ⊗ Ψ) ◦ (c12 ⊗id)(y i ⊗v i=1
= =
n X i=1 n X
δJ (yic )Sgi {(δJ ◦ c22 ⊗ id¯)(WB )} δJ (yic )δ¯J (¯ c22 (Sgi {WB }))
i=1 n X
= d¯12
! δX (yi )δ¯B (Sgi {WB }) ,
i=1
where gi = g · Ψ(vi ). Observe that f ◦ qN = (g ◦ qN ) · u and gi ◦ qN = (g ◦ qN ) · vi . We compute d¯12 [1A ⊗ Mf ◦qN ]δX (x) = [1J ⊗ Mf ◦qN ]δJ (xc ) = Sf ◦qN {[1J ⊗ WG ][δJ (xc ) ⊗ 1]} = Sf ◦q {(id ⊗ δG¯) ◦ δJ (xc )[1J ⊗ WG ]} N
= Sg◦qN {[1J ⊗ 1 ⊗ u](δJ ⊗ id¯) ◦ δJ (xc )[1J ⊗ WG ]} ˆ = Sg◦qN {(δJ ⊗ id) ◦ (c12 ⊗id)([1 A ⊗ u]δX (x))[1J ⊗ WG ]}, which is the limit of finite sums n X ˆ ˆ i )[1J ⊗ WG ]} Sg◦qN {(δJ ⊗ id) ◦ (c12 ⊗id)(y i ⊗v i=1
=
n X i=1
= d¯12
δJ (yic )S(g◦qN )·vi {1J ⊗ WG } n X i=1
! δX (yi )[1B ⊗ Mgi ◦qN ] .
Since d¯12 is an isometry we get the desired result. (ii) We write g ∗ = Ψ(u) · f for some u ∈ Cr∗ (G) and f ∈ A(G/N ). Here ∗ ˆ r∗ (G), it is the limit of finite sums gP(r) = g(r−1 ). Since δX (y)[1B ⊗ u] ∈ X ⊗C ˆ i xi ⊗vi . We compute d¯12 δX (y)δ¯B (Sg {WB })∗ = δJ (y c )δ¯J (¯ c22 (Sg∗ {W ∗ })) B
∗ ˆ ¯)([y ⊗1]W ˆ = Sf {(δJ ⊗ id¯) ◦ (c12 ⊗id B )[1J ⊗ Ψ(u)]} ˆ = Sf {(δJ ◦ c11 ⊗ id¯)(WA∗ )(δJ ⊗ Ψ) ◦ (c12 ⊗id)(δ X (y)[1B ⊗ u])},
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HUU HUNG BUI
which is the limit of finite sums n X ˆ ˆ i )} Sf {(δJ ◦ c11 ⊗ id¯)(WA∗ )(δJ ⊗ ΨN ) ◦ (c12 ⊗id)(x i ⊗v i=1
=
n X i=1
= d¯12
c11 (Sfi∗ {WA∗ }))δJ (xci ) δ¯J (¯ n X
! δ¯A (Sfi {WA })δX (xi ) ,
i=1
where fi = (Ψ(vi )·f )∗ . Observe that (g◦qN )∗ = u·(f ◦qN ) and (fi ◦qN )∗ = vi ·(f ◦qN ). We compute ∗ d¯12 δX (y)[1B ⊗ Mg◦qN ]∗ = δJ (y c )[1J ⊗ Mg◦q ] N
= S(g◦qN )∗ {[δJ (y c ) ⊗ 1][1J ⊗ WG∗ ]} ∗ c ¯ ∗ {[1J ⊗ W ](id ⊗ δG ) ◦ δJ (y )} = S (g◦qN )
= Sf ◦qN {[1J ⊗
G ∗ WG ](δJ
ˆ ⊗ id) ◦ (c12 ⊗id)(δ X (y)[1B ⊗ u])},
which is the limit of finite sums n X ˆ ˆ i )} Sf ◦qN {[1J ⊗ WG∗ ](δJ ⊗ id) ◦ (c12 ⊗id)(x i ⊗v i=1
=
n X i=1
Svi ·(f ◦qN ) {1J ⊗ WG∗ }δJ (xci )
= d¯12
n X i=1
! ∗
[1A ⊗ Mfi ◦qN ] δX (xi ) .
Note that in the Lemma 3, the proof of (ii) is very similar to that of (i). For convenience we have given both here. The arguments in the proof of Lemma 3 is also very similar to those of [B2, Proposition 1.3]. Proof of Theorem 2. Let X = X ×δX G denote the crossed product of Hilbert C ∗ module X as defined in [B2, Definition 1.2]. Set A = A ×δA G and B = B ×δB G. By [B2, Theorem 1.6], X is an A, B-imprimitivity bimodule. To prove the theorem, we need to show that IWA is the ideal of A corresponding to the ideal IWB of B via X in the sense of [R, Theorem 3.1]. It is enough to show that IWA X = X IWB . Let α, α0 ∈ A, f ∈ A(G/N ) and ξ ∈ X . By Lemma 3(i), nA (f )α0 ξ is the limit of finite sums of elements δX (y)nB (g)β 0 for all y ∈ X, g ∈ A(G/N ) and β 0 ∈ B. Since αδX (y) ∈ X for all y ∈ X, it follows that αnA (f )α0 ξ ∈ X IWB . Hence IWA X ⊂ X IWB . By a similar argument and Lemma 3(ii), we can show that ξβnB (g)∗ β 0 ∈ IWA X for all ξ ∈ X , β, β 0 ∈ B and g ∈ A(G/N ). Hence X IWB ⊂ IWA X . References [B] [B2]
H. H. Bui, Morita equivalence of twisted crossed products by coactions, J. Funct. Anal. 123 (1994), 59–98. MR 95g:46121 H. H. Bui, Crossed products of Hilbert C ∗ -modules, Proc. Amer. Math. Soc. (to appear). CMP 96:05
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S. Baaj and G. Skandalis, C ∗ alg` ebres de Hopf et th´ eorie de Kasparov ´ equivariante, KTheory 2 (1989), 683–721. MR 90j:46061 [BS2] S. Baaj and G. Skandalis, Unitaires multiplicatifs et dualite pour les produits cros´ es de ebres, Ann. Sci. Ec. Norm. Sup. 26 (1993), 425–488. MR 94e:46127 C ∗ -alg` [ER] S. Echterhoff and I. Raeburn, Multipliers of imprimitivity bimodules and Morita equivalence of crossed products, Math. Scand. 76 (1995), 289–309. CMP 96:02 [LPRS] M. B. Landstad, J. Phillips, I. Raeburn and C. E. Sutherland, Representations of crossed products by coactions and principal bundles, Trans. Amer. Math. Soc. 299 (1987), 747– 784. MR 88f:46127 [M] K. Mansfield, Induced representations of crossed products by coactions, J. Funct. Anal. 97 (1991), 112–161. MR 92h:46095 [PR] J. Phillips and I. Raeburn, Twisted crossed products by coactions, J. Austral. Math. Soc. (Series A) 56 (1994), 320–344. MR 95e:46079 [R] M. A. Rieffel, Unitary representations of group extension; an algebraic approach to the theory of Mackey and Blattner, Advances in Math., Suppl. Studies, vol. 4, Academic Press, New York, 1979, pp. 43–82. MR 81h:22004
[BS]
School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia E-mail address:
[email protected]
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