Fixed point property and the Fourier algebra of a locally compact group

Abstract

Abstract. We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) L 1 spaces and use this for the Fourier algebra A(G) of a locally compact group G. In particular we show that if G is an IN-group, then A(G) has the weak fpp if and only if G is compact. We also show that if G is any locally compact group, then A(G) has the fixed point property (fpp) if and only if G is finite. Furthermore if a nonzero closed ideal of A(G) has the fpp, then G must be discrete.

Fixed point properties of semigroups of nonexpansive mappings

“…It is clear that abelian semitopological semigroups and semitopological groups are left reversible. Other examples of left reversible semigroups and connections with the notion of left amenability can be found for instance in [10,[12][13][14]17,23]. For a topological space K, an action of S on K is a map ψ : S × K → K, denoted by ψ(s, k) = s(k) or sk for s ∈ S and k ∈ K, that satisfies s 1 s 2 (k) = s 1 (s 2 k) for all s 1 , s 2 ∈ S, and k ∈ K. We also assume that ψ is separately continuous, i.e., for every s 0 ∈ S and k 0 ∈ K, the maps s → sk 0 (S → K) and k → s 0 k (K → K) are continuous.…”

Fixed point properties of semigroups of non-expansive mappings

“…Kirk [21] extended this result by showing that if K is a weakly compact subset of E with normal structure, then K has the fixed point property. Other examples of Banach spaces with the weak fixed point property include c 0 , 1 , trace class operators on a Hilbert space and the Fourier algebra of a compact group (see [12,14,15,26,27,31,32,34,36,40] and [3,4] for more details). However, as shown by Alspach [1], L 1 [0, 1] does not have the weak fixed point property.…”

Weak⁎ fixed point property and asymptotic centre for the Fourier–Stieltjes algebra of a locally compact group

“…If G is unimodular and not an [AU] group, from [24] we obtain a still stronger statement: like above, but with "weak" in place of "weak * ". We do not know whether this stronger statement also holds for non-unimodular groups.…”