Invariant Means on Semigroups

Abstract

Let S be a semigroup, that is, a set endowed with an associative product

$$(s, t)\mapsto st.$$

We consider the (real) Banach space of all real-valued bounded functions on S, namely,

$$\ell ^\infty (S)=\Big \{f:S\rightarrow {\mathbb R}\,\,\,\text { such that}\,\,\,\Vert f\Vert \doteq \sup _{s\in S}|f(s)|<\infty \Big \}. $$

An element $f\in \ell ^\infty (S)$ is called positive if $f(s)\ge 0$ for every $s\in S$. A linear functional ${\Lambda }:\ell ^\infty (S)\rightarrow {\mathbb R}$ is called positive if ${\Lambda }f\ge 0$ for every positive element $f\in \ell ^\infty (S)$. We agree to denote a constant function on S by the value of the constant.