The Fixed Point Property of a Banach Algebra Generated by an Element with Infinite Spectrum. (10th June 2018) - Thongin, P.; Fupinwong, W.
Abstract : A Banach spaceX is said to have the fixed point property if for each nonexpansive mappingT : E → E on a bounded closed convex subsetE 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) ifx, y ∈ X is such thatτ x ≤ τ y, for eachτ ∈ Ω ( X ), thenx ≤ y, and (iii)inf ⁡ { r ( x ) : x ∈ X, x = 1 } > 0 . We prove that there exists an elementx 0 inX such that〈 x 0 〉 R = ∑ i = 1 k α i x 0 i : k ∈ N, α i ∈ R ¯ does not have the fixed point property. Moreover, as a consequence of the proof, we have that, for each elementx 0 inX with infinite spectrum andσ ( x 0 ) ⊂ R, the Banach algebra〈 x 0 〉 = ∑ i = 1 k α i x 0 i : k ∈ N, α i ∈ C ¯ generated byx 0 does not have the fixed point property.