IUMJ

Title: Universal and exotic generalized fixed-point algebras for weakly proper actions and duality

Authors: Alcides Buss and Siegfried Echterhoff

Issue: Volume 63 (2014), Issue 6, 1659-1701

Abstract:

Given a $C^{*}$-dynamical system $(A,G,\alpha)$, we say that $A$ is a weakly proper $X\rtimes G$-algebra if there exists a proper $G$-space $X$ together with a nondegenerate $G$-equivariant *-homomorphism $\phi:C_0(X)\to\mathcal{M}(A)$. Weakly proper $G$-algebras form a large subclass of the class of proper $G$-algebras in the sense of Rieffel. In this paper, we show that weakly proper $X\rtimes G$-algebras allow the construction of \emph{full} fixed-point algebras $A^{G,\alpha}_u$ corresponding to the full crossed product $A\rtimes_{\alpha}G$, thus solving, in this setting, a problem stated by Rieffel in his 1988 original article on proper actions. As an application, we obtain a general Landstad duality result for arbitrary coactions together with a new and functorial construction of maximalizations of coactions. The same methods also allow the construction of exotic generalized fixed-point algebras associated with crossed-product norms lying between the reduced and universal ones. Using these, we give complete answers to some questions on duality theory for exotic crossed products recently raised by Kaliszewski, Landstad, and Quigg.