A Banach spaceXis said to have the fixed point property if for each nonexpansive mappingT:E→Eon a bounded closed convex subsetEofXhas a fixed point. LetXbe an infinite dimensional unital Abelian complex Banach algebra satisfying the following: (i) condition (A) in Fupinwong and Dhompongsa, 2010, (ii) ifx,y∈Xis such thatτx≤τy,for eachτ∈Ω(X),thenx≤y,and (iii)inf{r(x):x∈X,x=1}>0.We prove that there exists an elementx0inXsuch 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 elementx0inXwith infinite spectrum andσ(x0)⊂R,the Banach algebra〈x0〉=∑i=1kαix0i:k∈N,αi∈C¯generated byx0does not have the fixed point property.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.