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

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_۰(X) is infinite dimensional, (iii) C_۰ (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