Topological semigroups and representations. I

Wong, James C. S.

doi:10.1090/S0002-9947-1974-0369604-8

Topological semigroups and representations. I
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by James C. S. Wong PDF

Trans. Amer. Math. Soc. 200 (1974), 89-109 Request permission

Abstract:

Let $S$ be a topological semigroup (separately continuous multiplication) with identity and $W(S)$ the Banach space of all weakly almost periodic functions on $S$. It is well known that if $S = G$ is a locally compact group, then $W(G)$ always has a (unique) invariant mean. In other words, there exists $m \in W{(G)^ \ast }$ such that $||m|| = m(1) = 1$ and $m({l_s}f) = m({r_s}f) = m(f)$ for any $s \in G,f \in W(G)$ where ${l_s}f(t) = f(st)$ and ${r_s}f(t) = f(ts),t \in S$ The main purpose of this paper is to present several characterisations (functional analytic and algebraic) of the existence of a left (right) invariant mean on $W(S)$ In particular, we prove that $W(S)$ has a left (right) invariant mean iff a certain compact topological semigroup $p{(S)^\omega }$ (to be defined) associated with $S$ contains a right (left) zero. Other results in this direction are also obtained.

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