Averaging almost-periodic functions and finite-dimensional unitary representations on free groups
type: Item
instance of: scholarly article
title: Averaging almost-periodic functions and finite-dimensional unitary representations on free groups
zbMATH Open document ID: 0705.43001
DOI: 10.1007/BF00972215
author: A. Z. Gorbis
author: A. A. Tempelman
published in: Lithuanian Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let G be a countable group. We say that a sequence \((A_ n)\) of finite sets in G is a right averaging sequence for a function \(\phi\) iff there is a number \(M(\phi)\) (right-side mean value) such that \(\lim_{n}| A_ n|^{-1}\sum_{g\in A_ n}\phi(fg)= M(\phi)\) uniformly in \(f\in G\). Obviously ``right'' can be replaced by ``left''. The sequence \((A_ n)\) is universally averaging for a linear translation invariant space of functions on G if it is right and left averaging for every function in that space. A typical case is that of almost periodic (a.p.) functions. It is not known if there exists a universal averaging sequence for a.p. functions in any countable group. Of course is does exist for amenable groups (owing to the Følner condition). In the free group \(F_ 2\) with two generators (a and b) Greenleaf has found universal averaging sequences for a.p. functions. He also has shown that some finite sets, which might be hoped to have this property do not have it (such is for example the sequence of sets \(E_ n=\{x_ 1...x_ n:\) \(x_ i=a\) or \(b\}\)). The authors solve the problem in the affirmative for all free groups \(F_ k\) with finitely many generators \(g_ 1,...,g_ k\). More exactly, they prove the following theorem, which implies such solution: Let \((a_ n)\) and \((b_ n)\) be sequences of natural numbers with \(a_ n<b_ n\), \(b_ n-a_ n\to \infty\) and let \(s^{(n)}_{a_ n}\), \(s^{(n)}_{a_ n+1},...,s_{b_ n}^{(n)}\) and \(t_{a_ n}^{(n)},t^{(n)}_{a_ n+1},...,t_{b_ n}^{(n)}\) be integers such that \(t_ i^{(n)}\geq s_ i^{(n)}\). Finally we put \[ A_ n=\{g\in F_ k:\;g=\prod^{b_ n}_{i=a_ n}g_{\bar i}^{j_ i},\quad s_ i^{(n)}\leq j_ i\leq t_ i^{(n)}-1\} \] where \(\bar {\i}\) means the residue of i mod k. If \[ \lim_{n}(b_ n-a_ n)^{- 1}\min \{t_ i^{(n)}-s_ i^{(n)}:\;a_ n\leq i\leq b_ n\}=\infty \] then, for every finite-dimensional unitary representation U of \(F_ k\), there exists \(\lim_{n}| A_ n|^{-1} \sum_{g\in A_ n}U(g)=P\) where P is the orthogonal projectio…
reviewed by: Stanisław Hartman
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 43A60
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 4157226
zbMATH Keywords: right averaging sequence
zbMATH Keywords: mean value
zbMATH Keywords: universally averaging
zbMATH Keywords: translation invariant space of functions
zbMATH Keywords: a.p. functions
zbMATH Keywords: amenable groups
zbMATH Keywords: free groups
zbMATH Keywords: finite-dimensional unitary representation
Wikidata QID: Q100163010
MaRDI profile type: Publication
cites work: Q5332414
cites work: Q3092574
cites work: Q5807151
cites work: Concrete methods for summing almost periodic functions and their relation to uniform distribution of semigroup actions
cites work: On the factorization of matrices
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description: scientific article; zbMATH DE number 4157226