Vector-valued invariant means revisited
type: Item
instance of: scholarly article
title: Vector-valued invariant means revisited
zbMATH Open document ID: 1027.46038
DOI: 10.1016/S0022-247X(02)00639-X
author: H. Bustos Domecq
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: In this paper, \(S\) is a set, \(Y\) is a quasi-Banach space and \(B(S, Y)\) the linear space of bounded functions from \(S\) into \(Y\) with the supremum norm. If \(S\) is a semigroup, for any \( f \in B(S, Y)\) and \( z \in S\), \(f_{z} \in B(S, Y) \) is defined by \(f_{z}(x)=f(x+z)\). If \(S\) is a directed set and \( f \in B(S, Y)\) and \(f_{0} \in Y\), then \( f=f_{0}\) eventually if there is a \( y \in S\) such that \( f(x) = f_{0}\) for each \(x \in S\) with \( x \geq y \). If \(S\) is a set and \(U\) an ultrafilter on \(S\), then the ultrapower of \(Y\), \(Y_{U}\), with respect to \(U\) is the quasi-Banach space obtained by taking the quotient of \( B(S, Y)\) by the subspace \(\{ f \in B(S, Y): \lim_{U(s)} \|f(s) \|_{Y} =0 \} \). For a semigroup \(S\), a right invariant average for \( B(S, Y)\) is a bounded linear operator \( m: B(S, Y) \to Y \) such that (i) \(m(f_{z}) =m(f)\), for every \( f \in B(S, Y)\) and \( z \in S \), and (ii) \(m(f) = f_{0}\) if \( f(x)=f_{0}\) for each \(x\). Left invariant averages can be defined in a similar way. For a directed set \(S\), a bounded linear operator \( a: B(S, Y) \to Y \) is called an admissible assignment if \(a(f)= f_{0}\) for any \(f=f_{0}\) eventually. A quasi-Banach space is called an ultrasummand if it is complemented in each ultrapower \(Y_{U}\). The main results proved in this paper are: I. (i) For an amenable semigroup \(S\) and a Banach space \(Y\), complemented in its bidual by a projection \(\pi\), \(B(S, Y)\) admits an invariant average of norm at most \( \|\pi \|\). (ii) Suppose \(Y\) is a Banach space such that, for every commutative semigroup \(S\), there is an invariant average of norm \( \leq K\). Then \(Y\) is complemented in its bidual by a projection of norm at most \(K\). (iii) For a Banach space the following are equivalent: (a) \(Y\) is complemented in its bidual. (b) For every commutative semigroup \(S\), there is an invariant average. (c) For every directed set \(S\), there is an admissibl…
reviewed by: Surjit Singh Khurana
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46A16
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 46B08
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 1866698
zbMATH Keywords: invariant average
zbMATH Keywords: admissible assignment
zbMATH Keywords: ultrapower
zbMATH Keywords: quasi-Banach space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866698