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Copyright © 2018 P.Thongin andW. Fupinwong.This is an open access article distributed under theCreativeCommonsAttribution License,whichpermits unrestricteduse, distribution, and reproduction in anymedium, provided the original work is properly cited. A Banach spaceX is said to have the fixed point property if for each nonexpansive mapping T : E → E on a bounded closed convex subset E ofX has a fixed point. LetX be an infinite dimensional unital Abelian complex Banach algebra satisfying the following: (i) condition (A) in Fupinwong and Dhompongsa, 2010, (ii) if x, y ∈ X is such that |τ(x)| ≤ |τ(y)|, for each τ ∈ Ω(X), then ‖x‖ ≤ ‖y‖, and (iii) inf{r(x) : x ∈ X, ‖x‖ = 1} > 0.We prove that there exists an element x0 inX such that ⟨x0⟩R = {∑ki=1 αixi0 : k ∈ N, αi ∈ R} does not have the fixed point property.Moreover, as a consequence of the proof, we have that, for each element x0 inXwith infinite spectrum and σ(x0) ⊂ R, the Banach algebra ⟨x0⟩ = {∑ki=1 αixi0 : k ∈ N, αi ∈ C} generated by x0 does not have the fixed point property.

The Fixed Point Property of a Banach Algebra Generated by an Element with Infinite Spectrum