Fixed point and spectral characterization of finite dimensional C*-algebras

TL;DR

This paper characterizes finite-dimensional C*-algebras through fixed point properties, spectral finiteness, and dimensionality, establishing their equivalence using spectral theory, projection properties, and fixed point theorems.

Contribution

It provides a new spectral and fixed point characterization of finite-dimensional C*-algebras, linking algebraic, spectral, and topological properties.

Findings

Finite-dimensional C*-algebras have the fixed point property for nonexpansive mappings.

Spectra of self-adjoint elements in such algebras are finite.

The three conditions are equivalent, characterizing finite-dimensionality.

Abstract

We show that the following conditions on a C*-algebra are equivalent: (i) it has the fixed point property for nonexpansive mappings, (ii) the spectrum of every self adjoint element is finite, (iii) it is finite dimensional. We prove that (i) implies (ii) using constructions given by Goebel, that (ii) implies (iii) using projection operator properties derived from the spectral and Gelfand-Naimark-Segal theorems, and observe that (iii) implies (i) by Brouwer's fixed point theorem.