Unique disintegration of arbitrary positive definite functions on \(*\)- divisible semigroups
type: Item
instance of: scholarly article
title: Unique disintegration of arbitrary positive definite functions on \(*\)- divisible semigroups
zbMATH Open document ID: 0739.43006
DOI: 10.1007/BF01160953
author: Paul Ressel
author: Torben Maack Bisgaard
published in: Mathematische Zeitschrift
publication date: 1992-06-25T00:00:00Z
full work available at URL: https://eudml.org/doc/174023
review text: Let \((S,*)\) denote an Abelian semigroup with involution and denote by \(S^*\) the character space of \(S\), i.e. the set of \(*\)-homomorphisms into the semigroup \(\mathbb{C}\) with conjugation as involution. It is known [\textit{R. J. Lindahl} and \textit{P. H. Maserick}, Duke Math. J. 38, 771-782 (1971; Zbl 0243.43004)] that every bounded positive definite function \(\varphi\) on \(S\) is the `Laplace' transform \(\hat\mu: s\mapsto\int_ S\sigma(s)d\mu(s)\) of a unique measure \(\mu\) on \(\hat S\), the space of bounded characters of \(S\). \textit{C. Berg}, \textit{J. P. R. Christenson} and \textit{P. Ressel} [Harmonic Analysis on semigroups (Springer 1984; Zbl 0619.43001)] call \(S\) perfect if every positive definite function \(\varphi\) is the Laplace transform of a unique positive Radon measure on \(S^*\). However there is no such Laplace transform for a positive function on \(\mathbb{R}_ +\). To be able to achieve this the authors use the Kolmogorov construction for measures rather than restricting themselves to positive Radon measures. Let \(\mathcal A\) be the smallest \(\sigma\)-algebra on \(S^*\) induced by the evaluation mapping \(\sigma\mapsto\sigma(s): S^*\to\mathbb{C}\). The authors permit measures \(\mu\) on \((S^*,\mathcal A)\) such that \(\sigma\mapsto\sigma(s)\) is \(\mu\)-integrable for all \(s\in S\), calling \(S\) quasi-perfect if each positive definite function is the Laplace transform of a unique such measure. For \(S\) countable, Borel measures are Radon so perfection is equivalent to quasiperfection. The authors show that quasiperfection is preserved by \(*\)-homomorphic images and products, as for perfection, and that it is preserved by arbitrary direct sums (as opposed to the corresponding result for direct sums of sequences of perfect spaces). They define a \(*\)-semigroup to be \(*\)- divisible if every \(s\in S\) can be written in the form \(s=mt+nt^*\), \(t\in S\), \(m+n\geq 2\). (For example an Abelian group \(G\) with \(s^*=-s\) …
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 3694
zbMATH Keywords: Abelian semigroup with involution
zbMATH Keywords: character space
zbMATH Keywords: bounded positive definite function
zbMATH Keywords: bounded characters
zbMATH Keywords: Laplace transform
zbMATH Keywords: positive Radon measure
zbMATH Keywords: Borel measures
zbMATH Keywords: perfection
zbMATH Keywords: quasiperfection
reviewed by: Aubrey Wulfsohn
MaRDI profile type: Publication
cites work: Q3755956
cites work: Extension of characters on *-semigroups
description: scientific article; zbMATH DE number 3694