Centralization of positive definite functions, weak containment of representations and Thoma characters for the infinite symmetric group
type: Item
instance of: scholarly article
title: Centralization of positive definite functions, weak containment of representations and Thoma characters for the infinite symmetric group
zbMATH Open document ID: 1120.22003
author: Takeshi Hirai
published in: Journal of Mathematics of Kyoto University
publication date: 2006-03-09T00:00:00Z
review text: For a discrete group \(G\) with identity \(e\), let \(P(G)\) denote the convex set of all positive definite functions \(f\) on \(G\) with \(f(e)=1\) and \(\text{Tr}(G)\) the subset of all traces on \(G\), that is, the conjugation invariant functions in \(P(G)\). The Gelfand-Naimark-Segal construction associates to each \(f\in P(G)\) a cyclic unitary representation \(\pi_f\) of \(G\), and \(\pi_f\) is irreducible if \(f\) is indecomposable (equivalently, an extreme point of \(P(G))\) and \(\pi_f\) is a factor representation if \(f\) is an extreme point of \(\text{Tr}(G)\) (a so-called character of \(G)\). In many cases, the characters can be described explicitly, whereas there is no way to determine the irreducible representations (up to unitary equivalence). For instance, the set of characters of the infinite symmetric group \(S_\infty= \cup^\infty_{n=1}S_n\) was determined by \textit{E. Thoma} [Math. Z. 85, 40--61 (1964; Zbl 0192.12404)] and has since attained a lot of attention. In the paper under review the author continues his studies of positive definite functions and unitary representations of \(S_\infty\). In order to provide a link between \(P(S_\infty)\) and \(\text{Tr}(S_\infty)\), he introduces the following centralization process. Let \((G_n)_n\) be an increasing sequence of finite subgroups of a group \(G\) with \(\bigcup^\infty_{n=1}G_n=G\). To any \(f\in P(G)\), associate the sequence \((f_n)_n\subseteq P(G)\) defined by \(f_n(g)=\frac {1}{|G_n|}\sum_{a \in G_n}f(a^{-1}ga)\), \(g\in G\), \(n\in\mathbb{N}\). If this sequence converges pointwise, then the limit is a trace. The main body of results of the paper consists of, partly fairly technical and intricate, exploitation of this centralization process. \textit{M. Bożejko} [Studia Math. 95, 107--118 (1989; Zbl 0714.43007)] and \textit{M. Bożejko} and \textit{R. Speicher} [Math. Ann. 300, 97--120 (1994; Zbl 0819.20043)] have shown that the following families of functions on \(S_\infty\) are contained i…
reviewed by: Eberhard Kaniuth
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 20C32
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A40
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 5010140
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250281694
OpenAlex ID: W1575196935
DOI: 10.1215/KJM/1250281694
description: scientific article; zbMATH DE number 5010140