The fixed point property of unital abelian Banach algebras (Q963583)

Let \(X\) be an infinite-dimensional unital abelian Banach algebra. It is shown that \(X\) does not have the fixed point property for nonexpansive mappings (fpp) if the character \(\tau\) on \(X\) and the spectral radius \(r(x)\) of \(x \in X\) satisfy the following conditions: (i) if \(|\tau(x)| \leq |\tau(y)|\), \(x,y \in X\), then \(||x|| \leq ||y||\); and (ii) inf \(\{r(x) : x \in X, ||x|| = 1 \} > 0\). Examples of algebras which fail the fpp are given, including algebras of continuous functions on compact sets.