Pull-Backs of C ∗ -Algebras and Crossed Products by Certain Diagonal Actions

Raeburn, Iain; Williams, Dana P.

doi:10.2307/1999675

Cited by 43 publications

(89 citation statements)

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“…see [23]) which carries canonical C 0 (V )-and C 0 (Y )-algebra structures). Recall from (1.9) that we have a C 0 (U )-linear (and hence also C 0 (X × X)-linear) isomorphism…”

Section: Remark 13 (1)mentioning

confidence: 99%

KK-theoretic duality for proper twisted actions

2007

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Abstract. Let A be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then A −1 ⋊G is KK-theoretically Poincaré dual to`A⊗ C 0 (X) Cτ (X)´⋊G, where A −1 is the inverse of A in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov's duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum-Connes conjecture.

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“…see [23]) which carries canonical C 0 (V )-and C 0 (Y )-algebra structures). Recall from (1.9) that we have a C 0 (U )-linear (and hence also C 0 (X × X)-linear) isomorphism…”

Section: Remark 13 (1)mentioning

confidence: 99%

KK-theoretic duality for proper twisted actions

2007

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“…There has nevertheless been considerable interest in situations where there is a useful analogue of A α in the multiplier algebra M(A) [5,8,23,30,33,36,37]. Here we are particularly interested in the proper actions introduced by Rieffel in [36]; in the motivating example, A = C 0 (T ) is commutative, G acts properly on the right of T , α is the action rt of G by right translation on functions, and the algebra C 0 (T /G), which we can view as a subalgebra of C b (T ) = M(C 0 (T )), is an excellent substitute for the missing fixed-point algebra.…”

Section: Introductionmentioning

confidence: 99%

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Huef

Kaliszewski

Raeburn

et al. 2009

Algebr Represent Theor

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Abstract. Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T . Rieffel proved that if α is an action of G on a C * -algebra A and there is an equivariant embedding of C 0 (T ) in M (A), then the action α of G on A is proper, and the crossed product A⋊ α,r G is Morita equivalent to a generalised fixed-point algebra Fix(A, α) in M (A) α . We show that the assignment (A, α) → Fix(A, α) extends to a functor Fix on a category of C * -dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel's Morita equivalence implements a natural isomorphism between a crossed-product functor and Fix. From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg-Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.

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“…Since G splits topologically we can write A >J α G as the iterated crossed product (A x αH H) x 7 L, where now 7 denotes the restriction of the canonical action of G on A x aH H to L. Since h ω defines an isomorphism between L and if, the calculations above show that this isomorphism carries the action, say 7, of L on (A x aH iί]Γonto the dual action of H on this space. The description of the isomorphism between (τes H )*(A) and Ax aiί H given in [25] (note that a H is locally unitary since G is compactly generated) shows that 7 is in fact the diagonal action Hence it is now a consequence of [24] that Now let us finally assume that S is nontrivial. Then there exists an ώ E …”

Section: Thus Since (P X V) Oj X Is Equivalent To (ρXf)o^ It Followmentioning

confidence: 99%