On the Schauder fixed point property. II. (English) Zbl 07758742

Salame, Khadime

Ann. Funct. Anal. 14, No. 4, Paper No. 78, 12 p. (2023).

Summary: The Schauder fixed point property \((\mathbf{F})\) was introduced and studied by Lau and Zhang as a semigroup formulation in the general setting of convex spaces of the well-known Schauder fixed point theorem in Banach spaces. What amenability property should possess a semigroup or a topological group to satisfy the Schauder fixed point property. Recently, the author provided a partial answer to that question and as a sequel, it is the purpose of this paper to study in more deep this problem. Our main result establishes that for a compact semitopological semigroup \(S\) we have: \(\mathrm{LUC}(S)\) is left amenable if, and only if, \(S\) has the fixed point property \((\mathbf{F})\). Furthermore, we also prove that totally bounded topological groups, semitopological groups \(S\) with the property that \(\mathrm{LUC}(S) \subset{\mathrm{aa}}(S)\), and strongly left amenable semitopological semigroups, possess all the Schauder fixed point property.
For Part I, see [K. Salame, Ann. Funct. Anal. 11, No. 1, 1–16 (2020; Zbl 1433.43003)].

MSC:

47H10	Fixed-point theorems
43A07	Means on groups, semigroups, etc.; amenable groups

Keywords:

almost (automorphic) periodic function; amenability; locally convex space; Schauder fixed point property; strongly amenable semigroup

Citations:

Zbl 1433.43003

References:

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