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In this paper, we deal with some fixed point properties for a
semi-topological semigroup S acting on a compact convex subset K of a Banach space. We first focus on the space LMC(S) of left multiplicatively continuous functions on S and its strong left amenability; the existence of a compact left ideal group in the
LMC-compactification of S. We then study the relation between left amenability and strong left amenability of LMC(S) with a common fixed point property for non-expansive and asymptotically non-expansive actions of S. Our results improve a result of T. Mitchell in 1970, and answer an open problem of A.T.-M. Lau in 2010 for the class of strongly left amenable semi-topological semigroups.
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