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

A normed space 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 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) c0(x) is infinite dimensional, (iii) c0(x) does not have the fixed 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 infinite, (ii) a is infinite dimensional, (iii) a does not have the fixed point property. moreover, we show that if a is an infinite dimensional complex c*–algebra (not necessarily commutative), then a does not have the fixed point property.