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

Abstract

© 2018 P. Thongin and W. Fupinwong. 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. Let X be an infinite dimensional unital Abelian complex Banach algebra satisfying the following: (i) condition (A) in Fupinwong and Dhompongsa, 2010, (ii) if x,yϵX is such that τx≤τy, for each τϵΩ(X), then x≤y, and (iii) inf{r(x):xϵX,x=1}>0. We prove that there exists an element x0 in X such that 〈x0〉R=i=1kμix0i:kϵN,μiϵR does not have the fixed point property. Moreover, as a consequence of the proof, we have that, for each element x0 in X with infinite spectrum and σ(x0)⊂R, the Banach algebra 〈x0〉=i=1kμix0i:kϵN,μiϵC generated by x0 does not have the fixed point property.