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

Abstract EN

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,y∈X is such that τx≤τy, 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 in X such that 〈x0〉R=∑i=1kαix0i: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 in X with infinite spectrum and σ(x0)⊂R, the Banach algebra 〈x0〉=∑i=1kαix0i:k∈N,αi∈C¯ generated by x0 does not have the fixed point property.