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Erschienen in:
2019 | OriginalPaper | Buchkapitel
23. Invariant Means on Semigroups
verfasst von : Vittorino Pata
Erschienen in: Fixed Point Theorems and Applications
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Abstract
Let S be a semigroup, that is, a set endowed with an associative product We consider the (real) Banach space of all real-valued bounded functions on S, namely, 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.
$$(s, t)\mapsto st.$$
$$\ell ^\infty (S)=\Big \{f:S\rightarrow {\mathbb R}\,\,\,\text { such that}\,\,\,\Vert f\Vert \doteq \sup _{s\in S}|f(s)|<\infty \Big \}. $$