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An Equivariant Theory for the Bivariant Cuntz Semigroup

Part of: $K$-theory and operator algebras Selfadjoint operator algebras Ordered structures

Published online by Cambridge University Press:  04 April 2018

Gabriele N. Tornetta
Gabriele N. Tornetta*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK ([email protected])
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Abstract

We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored, and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the one recently proposed by Gardella [‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math.43(3) (2017), 861–889], and we complement their work by providing an open projection picture of it.

Keywords

C*-algebrasgroup actionsCuntz semigroupcrossed productCuntz homology

MSC classification

Primary: 46L10: General theory of von Neumann algebras 46L35: Classifications of $C^*$-algebras
Secondary: 06F05: Ordered semigroups and monoids 19K14: $K_0$ as an ordered group, traces 46L30: States 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 573 - 598
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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