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# Vector-valued invariant means revisited

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## Abstract

We show that a Banach space X is complemented in its ultraproducts if and only if for every amenable semigroup S the space of bounded X-valued functions defined on S admits (a) an invariant average; or (b) what we shall call “an admissible assignment”. Condition (b) still provides an equivalence for quasi-Banach spaces, while condition (a) necessarily implies that the space is locally convex.

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... In order to prove that the converse holds true as well, in [2,4] the existence of an invariant mean on a certain idempotent semigroup S was assumed; the semigroup S comprises pairs (F, ε), where F ⊂ X * * is a finite-dimensional subspace and ε > 0; the semigroup operation was defined as (F, ε) · (G, δ) = (F + G, min{ε, δ}); related questions were also considered in [6]. ...
... S is the collection of non-empty multisets in F S , and F (2) S is the collection of non-empty non-singleton multisets in F S . ...
... is a subspace of W (note that the colexicographical ordering on F (2) S ensures that we have already defined g γ on all such subsets B); ...
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Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.
... Acknowledgement. We are indebted to Radosław Łukasik (Katowice) for having brought to our attention the problem of characterising Banach spaces which may be targets of invariant means on amenable semigroups and for pointing out Lipecki's Mathematical Review of Bustos Domecq's paper ( [5]), where a gap in the proof of [5, Theorem 2] was detected. We are also grateful to Wojciech Bielas (Prague) for spotting certain slips in the previous version of this note. ...
... so that P : X * * → κ X (X). That P is a linear projection and P C may be demonstrated exactly as in the proof of [5,Theorem 2]. ...
Article
Banach spaces that are complemented in the second dual are characterised precisely as those spaces $X$ which enjoy the property that for every amenable semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined on the Banach space of all bounded $X$-valued functions on $S$. This was first observed by Bustos Domecq (J. Math. Anal. Appl., 2002), however the original proof was slightly flawed as remarked by Lipecki. The primary aim of this note is to present a corrected version of the proof. We also demonstrate that universally separably injective spaces always admit invariant means with respect to countable amenable semigroups, thus such semigroups are not rich enough to capture complementation in the second dual as spaces falling into this class need not be complemented in the second dual.
... The existence of such invariant means for a fixed Banach space and for all amenable semigroups has been studied by Domecq [4,Theorem 1 and 2] and by the author in [12]. However, as observed by Lipecki in his Mathematical Review (MR1943762) of Bustos Domecq's paper, the proof of Theorem 2 contains a gap (a flawed choice of the semigroup, so we cannot use the Principle of Local Reflexivity). ...
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Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of X∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{**}$$\end{document}. We then show that X is complemented in X∗∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X^{**}$$\end{document} if and only if there exists an invariant mean M:ℓ∞(S,X)→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:\ell _\infty (S,X)\rightarrow X$$\end{document}. This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).
... In order to prove that the converse holds true as well, in [Bus02,Ka17] the existence of an invariant mean on a certain idempotent semigroup S was assumed; the semigroup S comprises pairs (F, ε), where F ⊂ X * * is a finitedimensional subspace and ε > 0; the semigroup operation was defined as (F, ε) · (G, δ) = (F + G, min{ε, δ}); related questions were also considered in [Lu17]. ...
Article
Full-text available
Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.
... Since X is complemented in its bidual we may then define (see [6]) for any bounded f : ...
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We present new methods to obtain singular twisted sums $X\oplus_\Omega X$ (i.e., exact sequences $0\to X\to X\oplus_\Omega X \to X\to 0$ in which the quotient map is strictly singular), in which $X$ is the interpolation space arising from a complex interpolation scheme and $\Omega$ is the induced centralizer. Although our methods are quite general, in our applications we are mainly concerned with the choice of $X$ as either a Hilbert space, or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces, including the only known example so far: the Kalton-Peck space $Z_2$. In the second case we obtain the first example of an H.I. twisted sum of an H.I. space. We then use Rochberg's description of iterated twisted sums to show that there is a sequence $\mathcal F_n$ of H.I. spaces so that $\mathcal F_{m+n}$ is a singular twisted sum of $\mathcal F_m$ and $\mathcal F_n$, while for $l>n$ the direct sum $\mathcal F_n \oplus \mathcal F_{l+m}$ is a nontrivial twisted sum of $\mathcal F_l$ and $\mathcal F_{m+n}$. We also introduce and study the notion of disjoint singular twisted sum of K\"othe function spaces and construct several examples involving reflexive $p$-convex K\"othe function spaces, which include the function version of the Kalton-Peck space $Z_2$.
... The existence of an invariant mean in the space of weakly almost periodic functions was investigated by de-Leew and Glicksberg [14], [15]. Vector-valued invariant means have been used by a number of authors for the study of some vectorvalued function spaces, functional equations , a linear topological classification of spaces of continuous functions and for solving stability problems [1], [2], [6], [23]- [25]. ...
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A definition of an invariant averaging for a linear representation of a group in a locally convex space is given. Main results: A group $H$ is finite if and only if every linear representation of $H$ in a locally convex space has an invariant averaging. A group $H$ is amenable if and only if every almost periodic representation of $H$ in a quasi-complete locally convex space has an invariant averaging. A locally compact group $H$ is compact if and only if every strongly continuous linear representation of $H$ in a quasi-complete locally convex space has an invariant averaging.
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Main results: For every equicontinuous almost periodic linear representation of a group in a complete locally convex space L with the countability property, there exists the unique invariant averaging; it is continuous and is expressed by using the L-valued invariant mean of Bochner and von-Neumann. An analog of Wiener's approximation theorem for an equicontinuous almost periodic linear representation in a locally convex space with the countability property is proved.
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