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

Szabó, Gábor

Canadian mathematical bulletin, Tome 58 (2015) no. 2, pp. 374-380

Résumé

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 ${{A}^{\alpha }}$ .

Détail

DOI : 10.4153/CMB-2014-074-x

Mots-clés : 46L55, 19K35, Rokhlin property, UCT, KK-theory, E-theory, circle actions