## Topological Methods in Nonlinear Analysis

### Amenability and Hahn-Banach extension property for set valued mappings

#### Abstract

Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterizations of a semitopological semigroup $S$ on the existence of a left invariant mean on ${\rm LUC}(S)$, ${\rm AP}(S)$ and ${\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of the Banach algebras related to amenability of groups and semigroups.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 2 (2019), 547-573.

Dates
First available in Project Euclid: 10 May 2019

https://projecteuclid.org/euclid.tmna/1557453832

Digital Object Identifier
doi:10.12775/TMNA.2019.011

Mathematical Reviews number (MathSciNet)
MR3983985

#### Citation

Lau, Anthony To-Ming; Yao, Liangjin. Amenability and Hahn-Banach extension property for set valued mappings. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 547--573. doi:10.12775/TMNA.2019.011. https://projecteuclid.org/euclid.tmna/1557453832

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