Fixed point property for Banach algebras associated to locally compact groups
Let \(E\) be a Banach space and \(K\) be a nonempty bounded convex subset of \(E\). We say that \(K\) has the fixed point property if every nonexpansive mapping \(T: K\mapsto K\) (i.e., \(\|Tx-Ty\|\leq \|x-y\|\) for all \(x,y\in K\)) has a fixed point. We say that \(E\) has the weak fixed point property if every weakly compact convex subset of \(E\) has the fixed point property. A dual Banach space \(E\) is said to have the weak* fixed point property if each weak* compact convex subset of \(E\) has the fixed point property. Let \(S\) be a semitopological semigroup, \(S\) is called left reversible if \(\overline{aS}\cap\overline{bS}\neq\empty\) for any \(a,b\in S\), where \(\overline{K}\) denotes the closure of \(K\). Clearly, abelian semigroups and groups are left reversible. We say that a Banach space \(E\) has the weak fixed point property for left reversible semigroups if whenever \(S\) is a left reversible semitopological semigroup, \(K\) is a nonempty weakly compact convex subset of \(E\), and \(S\) acts on \(K\) so that the action is separately continuous and nonexpansive, then \(K\) has a common fixed point for \(S\). Similarly we can define the weak* fixed point property for left reversible semigroups. In the paper under review the authors investigate the question under which conditions various Banach algebras associated to a locally compact group \(G\) have the weak* fixed point property for left reversible semigroups. They prove for example, that if \(G\) is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier-Stieltjes algebra of \(G\) has the weak* fixed point property for left reversible semigroups if and only if \(G\) is compact. This generalizes a classical result of T.~C.~Lim for the case when \(G\) is the circle group.
scientific article; zbMATH DE number 5655682
scientific article