The fixed point property of a non-unital abelian banach algebra generated by an element with infinite spectrum

Abstract

© 2020 by TJM. All rights reserved. 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. Assume that X is an infinite dimensional non-unital Abelian Banach algebra satisfying: (i) condition (A) defined in [W. Fupinwong, S. Dhompongsa, The fixed point property of unital Abelian Banach algebras, Fixed Point Theory and Applications (2020)], (ii) ‖x‖ ≤ ‖y‖ for each x, y ∈ X such that |τ(x)| ≤ |τ(y)| for each τ ∈ Ω(X), (iii) inf{r(x): x ∈ X, ‖x‖ = 1} > 0. We show that there is an element (x0, 0) in X such that [formula presented] does not have the fixed point property. This result is a generalization of Theorem 21 in [P. Thongin, W. Fupinwong, The fixed point property of a Banach algebra generated by an element with infinite spectrum, Journal of Function Spaces (2018)]. Moreover, as a consequence of the proof, for each element (x0, 0) in X with infinite spectrum and σ(x0, 0) ⊂ R, the Banach algebra generated by (x0, 0) [formula presented] does not have the fixed point property.