C*-algebraic approach to fixed point theory generalizes Baggett's theorem to groups with discrete reduced duals

TL;DR

This paper extends fixed point theory in the context of C*-algebras and locally compact groups, proving new theorems about group compactness and fixed point properties, and providing counterexamples and variants of existing theorems.

Contribution

It generalizes Baggett's theorem to groups with discrete reduced duals using C*-algebraic methods and addresses open problems in fixed point theory and group duality.

Findings

Second countable groups with discrete reduced duals are compact.

Constructed a scattered C*-algebra whose dual lacks weak* fixed point property.

Proved a variant of Bruck's fixed point theorem for von Neumann algebra preduals.

Abstract

In this paper, we show that if the reduced Fourier-Stieltjes algebra $B_{\rho}(G)$ of a second countable locally compact group $G$ has either weak* fixed point property or asymptotic center property, then $G$ is compact. As a result, we give affirmative answers to open problems raised by Fendler and et al. in 2013. We then conclude that a second countable group with a discrete reduced dual must be compact. This generalizes a theorem of Baggett. We also construct a compact scattered Hausdorff space $\Omega$ for which the dual of the scattered C*-algebra $C(\Omega)$ lacks weak* fixed point property. The C*-algebra $C(\Omega)$ provides a negative answer to a question of Randrianantoanina in 2010. In addition, we prove a variant of Bruck's generalized fixed point theorem for the preduals of von Neumann algebras. Furthermore, we give some examples emphasizing that the conditions in Bruck's generalized conjecture (BGC) can not be weakened any more.