Fixed point property for Banach algebras associated to locally compact groups

Abstract

In this paper we investigate when various Banach spaces associated to a locally compact group G have the fixed point property for nonex-pansive mappings or normal structure. We give sufficient conditions and some necessary conditions about G for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a C *-algebra A has the fixed point property then for any normal element a of A, the spectrum σ(a) is countable and that the group C *-algebra C * (G) has weak normal structure if and only if G is finite.

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

“…We give a more elementary proof of this fact and also show that this property characterizes compact groups among all locally compact groups. This answers a problem raised in Remark 4.3 [37] and improves a result of Lau and Mah [27] for separable compact groups. This paper is organized as follows: In Section 3 we show for the classical example of a non-compact [AU]-group, the Fell group F , that there is a non-empty weak * -compact convex subset K of B(F ) and an isometry T from K into K which is fixed point free.…”

“…This answers two open problems (Problem 8 and Problem 9) posed in the 1989 conference "Fixed point theory and applications" held in Marseille-Luminy [23] (see also Open Problem 6.6 in [27]). …”

“…It was proved in [27] for sequences. Note that in consideration of nets, we have to avoid the diagonalization process employed in [34] and [27]. Since this is not entirely obvious we give some details of the proof.…”

“…Lim when G is the circle group [34], and for separable compact groups by Lau and Mah [27]. We begin with the following lemma.…”

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.…”