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Regularity of the Semigroup of Regular Probability Measures on Locally Compact Hausdorff Topological Groups in which every element is of finite order

M N N Namboodiri

Abstract

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular elements in certain subsemigroups of P(G) (Theorem 4.1 [11]) for compact G remains true for locally compact G. In addition, a complete description of algebraically regular elements in P(G) has been established when G is countable or uncountable where every proper subgroup is countable. In this case the standing assumption that every element is of finite order is not required. For compact Lie groups, Fourier transform techniques are also used to get more information on P(G). Several concrete examples are provided to illustrate the observations.

Regularity of the Semigroup of Regular Probability Measures on Locally Compact Hausdorff Topological Groups in which every element is of finite order

Abstract

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular elements in certain subsemigroups of P(G) (Theorem 4.1 [11]) for compact G remains true for locally compact G. In addition, a complete description of algebraically regular elements in P(G) has been established when G is countable or uncountable where every proper subgroup is countable. In this case the standing assumption that every element is of finite order is not required. For compact Lie groups, Fourier transform techniques are also used to get more information on P(G). Several concrete examples are provided to illustrate the observations.

Paper Structure

This paper contains 8 sections, 11 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric Characterization of Generalised Invertible Elements in $P(\mathcal{G})$
  4. Jonsson-Type Groups
  5. Non commutative Fourier Transform
  6. Minimal Regular Semigroups Containing $P(\mathcal{G})$
  7. Moore-Penrose of Probability Measures
  8. Declarations

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a compact Hausdorff topological group such that every element in $\mathcal{G}$ is of finite order. Let $M(\mathcal{G}) = \text{Conv}\{\delta_{g}: g \in \mathcal{G}\}$, where 'Conv' denotes the convex hull. For an arbitrary finite set $\{g_{1}, g_{2}, ..., g_{n}\} \subset G$, the is regular in $M(\mathcal{G})$.Moreover, the elements of the following form namely, are also regul

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.2
  • ...and 18 more