Every invariant measure semigroup contains an ideal which is embeddable in a group
An abelian Hausdorff topological semigroup \(S\) is called an invariant measure semigroup if it is endowed with an inner regular measure \(\mu\) on the sigma ring generated by the compact subsets \({\mathcal C}(S)\) of \(S\) such that \(\mu(xC) = \mu(C)\) holds for each compact subset \(C \subseteq S\) and \(x \in S\), and there exists a neighborhood basis of subsets \(U \subseteq S\) satisfying \[ \mu_*(U) := \sup \{ \mu(C) : C \in {\mathcal C}(S), C \subseteq U\} < \infty. \] In the paper under review it is shown that every invariant measure semigroup \(S\) contains an ideal \(S_0\) such that \(S_0\) can be embedded as an open subsemigroup into a locally compact abelian group \(G\) in such a way that the restriction of \(\mu\) to \(S_0\) extends to a Haar measure on \(G\). Here the group \(G\) is obtained by first describing a regular representation \(\tau\) of \(S\) on the Hilbert space \(L^2(S,\mu)\) which turns out to be a strongly continuous unitary representation, and then to define \(G := \tau(S)\tau(S)^{-1}\). As a consequence of the embedding \(S_0 \to G\), a generalization of Pontryagin duality is obtained: the group \(G\) is isomorphic to the character group of the group of unitary characters of \(S\).
scientific article; zbMATH DE number 1356240