Vector-valued invariant means revisited
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 admissible assignment in \(B(S, Y)\). II. Suppose \(Y\) is a quasi-Banach space such that \( B({\mathbb Z}, Y)\) admits an invariant average. Then \(Y\) is isomorphic to a Banach space (here \({\mathbb Z}\) denotes the integers). III. A quasi-Banach space is an ultrasummand if and only if for every directed set \(S\), the space \(B(S, Y)\) has admissible assignments.
scientific article; zbMATH DE number 1866698