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A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in E-theory

Published online by Cambridge University Press:  20 November 2018

Gábor Szabó
Affiliation:
Westfälische Wilhelms-Universität, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany. e-mail: gabor.szabo@uni-muenster.de
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Abstract

Let $G$ be a metrizable compact group, $A$ a separable ${{\text{C}}^{*}}$ -algebra, and $\alpha :G\,\to \,\text{Aut}\left( A \right)$ a strongly continuous action. Provided that $\alpha $ satisfies the continuous Rokhlin property, we show that the property of satisfying the $\text{UCT}$ in $E$ -theory passes from $A$ to the crossed product ${{\text{C}}^{*}}$ -algebra $\mathcal{A}{{\rtimes }_{\alpha }}\,G$ and the fixed point algebra ${{A}^{\alpha }}$ . This extends a similar result by Gardella for $KK$ -theory in the case of unital ${{\text{C}}^{*}}$ -algebras but with a shorter and less technical proof. For circle actions on separable unital ${{\text{C}}^{*}}$ -algebras with the continuous Rokhlin property, we establish a connection between the $E$ -theory equivalence class of $A$ and that of its fixed point algebra ${{A}^{\alpha }}$ .

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

References

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