Document Type : Research Paper

Author

• Davood Alimohammadi

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 $T:E⟶E$ 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 ${\mathsf{C}}^{\ast }$-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 ${\mathsf{C}}^{\ast }$-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