Nonexpansive mappings on complex C?-algebras and their xed points

A normed space X is said to have the xed point property, if for each nonexpansive mapping T : E - E on a nonempty bounded closed convex subset E of X has a xed point. In this paper, we rst show that if X is a locally compact Hausdorff space then the following are equivalent: (i) X is innite set, (ii) C0(X) is innite dimensional, (iii) C0(X) does not have the xed point property. We also show that if A is a commutative complex C?{algebra with nonempty carrier space, then the following statements are equivalent: (i) Carrier space of A is innite, (ii) A is innite dimensional, (iii) A does not have the xed point property. Moreover, we show that if A is an innite dimensional complex C?{algebra (not necessarily commutative), then A does not have the xed point property.