Nonexpansive mappings on Abelian Banach algebras and their fixed points

A Banach space X is said to have the fixed point property if for each nonexpansive mapping T:E→E on a bounded closed convex subset E of X has a fixed point. We show that each infinite dimensional Abelian complex Banach algebra X satisfying: (i) property (A) defined in (Fupinwong and Dhompongsa in Fixed Point Theory Appl. 2010:Article ID 34959, 2010), (ii) ∥x∥≤∥y∥ for each x,y∈X such that |τ(x)|≤|τ(y)| for each τ∈Ω(X), (iii) inf{r(x):x∈X,∥x∥=1}>0 does not have the fixed point property. This result is a generalization of Theorem 4.3 in (Fupinwong and Dhompongsa in Fixed Point Theory Appl. 2010:Article ID 34959, 2010).