Fixed point property of real unital abelian banach algebras and their closed subalgebras 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. Let X be an infinite dimensional unital Abelian real Banach algebra with Ω(X) ≠ ∅ satisfying: (i) if x, y ∈ X is such that |τ(x)| ≤ |τ(y)|, for each τ ∈Ω(X), then ‖x‖ ≤ ‖y‖, (ii) inf{rX (x): x ∈ X, ‖x‖ = 1} > 0. We prove that, for each element x0 in X with infinite spectrum, the Banach algebra [formula presented] generated by x0 does not have the fixed point property.