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

Alimohammadi, Davood

doi:10.22075/ijnaa.2015.289

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

Document Type : Research Paper

Author

Davood Alimohammadi

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

10.22075/ijnaa.2015.289

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 $\mathsf{C}^*$-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 $\mathsf{C}^*$-algebra (not necessarily commutative), then $ A $ does not have the fixed point property.

Keywords

Banach space

C*-algebra

Fixed point property

Nonexpansive mapping

normed linear space