Nonlinear submeans on semigroups. (English) Zbl 1039.43002

Lau, Anthony To-Ming; Takahashi, Wataru

Topol. Methods Nonlinear Anal. 22, No. 2, 345-353 (2003).

Let \(S\) be a semigroup and \(X\) be a subspace of \(l^\infty(S)\) containing constants, where \(l^\infty(S)\) denotes the Banach space of bounded real-valued functions on \(S\) with supremum norm. A continuous linear functional \(\mu\) on \(X\) is called a {mean} if \(\| \mu\| =\mu(1)=1\). A real-valued function \(\mu\) on \(X\) is called a (nonlinear) {submean} if the following conditions hold: (1) \(\mu(f+g)\leq\mu(f)+\mu(g)\) for every \(f, g\in X\); (2) \(\mu(\alpha f)=\alpha\,\mu(f)\) for every \(f\in X\) and \(\alpha\geq 0\); (3) for \(f, g\in X\), \(f\leq g\) implies \(\mu(f)\leq\mu(g)\); (4) \(\mu(c)=c\) for every constant function \(c\). The purpose of the paper is to study some algebraic structure of submeans of certain spaces \(X\) and to find local conditions on \(X\) in terms of submean for the existence of a left invariant mean.

Reviewer: Sergei Platonov (Petrozavodsk)

MSC:

43A07	Means on groups, semigroups, etc.; amenable groups
47H20	Semigroups of nonlinear operators

Keywords:

invariant means; submean; left reversible semigroup