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Fixed Point Theorems for Measurable Semigroups of Operations

Published online by Cambridge University Press:  20 November 2018

James C. S. Wong*
Affiliation:
Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4
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Abstract

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Let Sbe a topological semigroup, K a compact convex subset of a separated convex space Eand T: S x KK an affine action (denoted by (s, x)Ts(x),sS, xK) of S as continuous affine maps on K. It is shown in A. Lau and J. Wong [22] that the weakly left uniformly measurable functions WLUM(S) on S has a left invariant mean iff Shas the fixed point property for weakly measurable affine actions, i.e. affine actions such that the scalar function sx*Ts(x) is measurable for each xK and x*E* (the dual of E) with respect to the Borel sets in S. It is natural to ask for a “strongly” measurable analogue of this result. There are a number of ways to define such actions and the corresponding functions on S. In this paper, we obtained a neat analogue of this fixed point theorem by a suitable choice of strong measurability which naturally leads to another new fixed point theorem for separable actions. Also, we shall unify these and many known fixed point theorems and extend and generalise them to anti-actions of S as bounded linear operators on Banach spaces

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1992

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