Every invariant measure semigroup contains an ideal which is embeddable in a group
type: Item
instance of: scholarly article
title: Every invariant measure semigroup contains an ideal which is embeddable in a group
zbMATH Open document ID: 0933.22006
author: A. R. Mirotin
published in: Semigroup Forum
publication date: 2000-03-27T00:00:00Z
review text: 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 \(S\).
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 22A20
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 43A10
zbMATH DE Number: 1356240
zbMATH Keywords: Pontryagin duality
zbMATH Keywords: measure semigroup
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: semigroup ideal
zbMATH Keywords: shift operator
reviewed by: Karl-Hermann Neeb
MaRDI profile type: Publication
Recommended article: Q3726369
Recommended article: Q3722743
Recommended article: Q4228148
Recommended article: Invariant measures on topological semigroups which have an ideal with open translation mappings
Recommended article: The Existence of Universal Invariant Semiregular Measures on Groups
Recommended article: A sufficient condition on embedding semigroups in groups
Recommended article: The semigroup of continuous selfmaps ofIhas infinitely many ideals
Recommended article: Every Commutative Semigroup Can be Embedded into a Complementary Semigroup with Involution
Recommended article: Every finite semigroup is embeddable in a finite relatively free semigroup.
Recommended article: Q6038809
description: scientific article; zbMATH DE number 1356240