A normed space X is said to have the xed point property, if for each nonexpansive mapping : E!Eon a nonempty bounded closed convex subset Eof X has a xed point. In this paper,e rst show that if Xis a locally compact Hausdor space then the following are equivalent: (i) Xin nite set, (ii) C 0 (X) is in nite dimensional, (iii) C (X) does not have the xed point property. We also show that if Ais a commutative complex C ? 0 {algebra with nonempty carrier space, then the following statements are equivalent: (i) Carrier space of Ais in nite, (ii) Ais in nite dimensional, (iii) Adoes not have the xed point property. Moreover, we show that if Ais an in nite complex ? {algebra (not necessarily commutative), then Adoes not have the xed point property.
