# Nonexpansive mappings on complex C*-algebras and their fixed points

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Science, Arak university, Arak 38156-8-8349, Iran

Abstract

A normed space $mathfrak{X}$ is said to have the fixed point property, if for each nonexpansive mapping $T : E longrightarrow E$ on a nonempty bounded closed convex subset $E$ of $mathfrak{X}$ has a fixed point. In this
paper, we first show that if $X$ is a locally compact Hausdorff space then the following are equivalent: (i) $X$ is infinite set, (ii) $C_0(X)$ is infinite dimensional, (iii) $C_0 (X)$ does not have the fixed point property. We also show that if $A$ is a commutative complex $mathsf{C}^star$--algebra with nonempty carrier space, then the following statements are equivalent: (i) Carrier space of $A$ is infinite, (ii) $A$ is infinite dimensional, (iii) $A$ does not have the fixed point property. Moreover, we show that if $A$ is an infinite complex $mathsf{C}^star$--algebra (not necessarily commutative), then $A$ does not have the fixed point property.

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