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

TL;DR

This paper develops a comprehensive theory of fixed-point algebras for weakly proper G-algebras, extending duality and maximalization results to exotic crossed-product norms, thus advancing the understanding of proper actions in C*-dynamics.

Contribution

It introduces a new framework for constructing fixed-point algebras for weakly proper actions, solving a longstanding problem and extending duality theory to exotic crossed products.

Findings

Constructed full fixed-point algebras for weakly proper G-algebras.

Established a general Landstad duality for arbitrary coactions.

Provided answers to duality questions for exotic crossed products.

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)->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 full fixed-point algebras A^G corresponding to the full crossed product A\rtimes_{\alpha}G, thus solving, in this setting, a problem stated by Rieffel in his 1988's 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 to 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.