Naturality and induced representations

Echterhoff, Siegfried; Kaliszewski, S.; Quigg, John; Raeburn, Iain

doi:10.1017/s0004972700022449

Cited by 38 publications

(78 citation statements)

References 15 publications

(22 reference statements)

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“…It is shown in [2,Proposition 3.3] that there is a category C*act(G) whose objects are dynamical systems (A, α) consisting of an action α of G on a C * -algebra A, and whose morphisms from (A, α) to (B, β) are isomorphism classes [X, u] of rightHilbert A -B bimodules X which carry an α-β compatible action u of G satisfying…”

Section: The Semi-comma Categorymentioning

confidence: 99%

“…In the semicomma category C*act(G, (C 0 (T ), rt)), the objects (A, α, φ A ) are systems (A, α) in C*act(G) together with a nondegenerate homomorphism φ A : C 0 (T ) → M(A) which is rt -α equivariant, and the morphisms from (A, α, φ A ) to (B, β, φ B ) are just the usual morphisms [X, u] from (A, α) to (B, β) in C*act(G), with the same composition defined by balanced tensor product of right-Hilbert bimodules. It follows immediately from [2,Proposition 3.3] that the semi-comma category is indeed a category. This may seem an unusual choice of category, and we will say more at the end of the section about our reasons for choosing it (see Remark 2.4).…”

Section: The Semi-comma Categorymentioning

confidence: 99%

“…The objects in our underlying category C* are C * -algebras, and the morphisms from A to B are isomorphism classes of the rightHilbert A -B bimodules which were invented by Rieffel [35] to place the theory of induced representations of groups in a C * -algebraic setting. The category C* was introduced in [2] and [3] as a setting for imprimitivity theorems for crossed products by actions and coactions, and was independently discovered in other contexts by Landsman [18,19] and by Schweizer [38]. It has the attractive feature that the invertible morphisms are those which are based on imprimitivity bimodules (see, for example, [2, Proposition 2.6]).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Semi-comma Categorymentioning

confidence: 99%

Section: The Semi-comma Categorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Naturality of Rieffel’s Morita Equivalence for Proper Actions

Huef

Kaliszewski

Raeburn

et al. 2009

Algebr Represent Theor

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“…We also note that property (iii) implies that each v(g) is an isometry on X, and that v is simply a unitary representation of G on the Hilbert space X when A = C. Equivariant representations of Σ may alternatively be described via (α, σ)-(α, σ) compatible actions of G on C * -correspondences over A, in the spirit of [18,19] (see also Remark 5.1).…”

Section: Equivariant Representationsmentioning

confidence: 99%

The Fourier–Stieltjes algebra of a C*-dynamical system

Bédos

Conti

2016

Int. J. Math.

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In analogy with the Fourier-Stieltjes algebra of a group, we associate to a unital discrete twisted C * -dynamical system a Banach algebra whose elements are coefficients of equivariant representations of the system. Building upon our previous work, we show that this Fourier-Stieltjes algebra embeds continuously in the Banach algebra of completely bounded multipliers of the (reduced or full) C * -crossed product of the system. We introduce a notion of positive definiteness and prove a Gelfand-Raikov type theorem allowing us to describe the Fourier-Stieltjes algebra of a system in a more intrinsic way. We also propose a definition of amenability for C * -dynamical systems and show that it implies regularity. After a study of some natural commutative subalgebras, we end with a characterization of the Fourier-Stieltjes algebra involving C * -correspondences over the (reduced or full) C * -crossed product.MSC 2010: 46L55, 43A50, 43A55.

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“…The objects are still G-algebras, but the morphisms are equivariant Hilbert bimodules. We refer to Section 6 of [Ech10] and to [EKQR00] for more details.…”

Section: The Induced Action Indmentioning

confidence: 99%

Semiprojectivity with and without a group action

Phillips

Sørensen

Thiel

2015

Journal of Functional Analysis

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Abstract. The equivariant version of semiprojectivity was recently introduced by the first author. We study properties of this notion, in particular its relation to ordinary semiprojectivity of the crossed product and of the algebra itself.We show that equivariant semiprojectivity is preserved when the action is restricted to a cocompact subgroup. Thus, if a second countable compact group acts semiprojectively on a C * -algebra A, then A must be semiprojective. This fails for noncompact groups: we construct a semiprojective action of Z on a nonsemiprojective C * -algebra.We also study equivariant projectivity and obtain analogous results, however with fewer restrictions on the subgroup. For example, if a discrete group acts projectively on a C * -algebra A, then A must be projective. This is in contrast to the semiprojective case.We show that the crossed product by a semiprojective action of a finite group on a unital C * -algebra is a semiprojective C * -algebra. We give examples to show that this does not generalize to all compact groups.Equivariant semiprojectivity was introduced in [Phi12], by applying the usual definition of semiprojectivity to the category of unital G-algebras (C * -algebras with actions of the group G) with unital G-equivariant * -homomorphisms. See Definition 1.1 below. The purpose of [Phi12] was to show that certain actions of compact groups on various specific C * -algebras are semiprojective. In particular, it is shown that any action of a second countable compact group on a finite dimensional C * -algebra is semiprojective, and that for n < ∞, quasifree actions of second countable compact groups on the Cuntz algebras O n are semiprojective.In this paper we study equivariant semiprojectivity more abstractly. We also introduce equivariant projectivity and carry out a parallel study of it. We extend the definition to allow actions by general locally compact groups, and we consider the nonunital version of equivariant semiprojectivity.From the work in [Phi12], it is not even clear whether a semiprojective action of a noncompact group can exist. One reason for skepticism was that the trivial

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Naturality and induced representations

Cited by 38 publications

References 15 publications

Naturality of Rieffel’s Morita Equivalence for Proper Actions

Naturality of Rieffel’s Morita Equivalence for Proper Actions

The Fourier–Stieltjes algebra of a C*-dynamical system

Semiprojectivity with and without a group action

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