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

  Published:2015-03-03
 Printed: Jun 2015
  • Gábor Szabó,
    Westfälische Wilhelms-Universität, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
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Abstract

Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$-algebra and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action. Provided that $\alpha$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in $E$-theory passes from $A$ to the crossed product $\mathrm{C}^*$-algebra $A\rtimes_\alpha G$ and the fixed point algebra $A^\alpha$. This extends a similar result by Gardella for $KK$-theory in the case of unital $\mathrm{C}^*$-algebras, but with a shorter and less technical proof. For circle actions on separable, unital $\mathrm{C}^*$-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 $A^\alpha$.
Keywords: Rokhlin property, UCT, KK-theory, E-theory, circle actions Rokhlin property, UCT, KK-theory, E-theory, circle actions
MSC Classifications: 46L55, 19K35 show english descriptions Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Kasparov theory ($KK$-theory) [See also 58J22]
46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
19K35 - Kasparov theory ($KK$-theory) [See also 58J22]
 

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