Nonlinear submeans on semigroups (Q1428725)

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.