arXiv subjects

Jorge Galindo

Publications and source records attributed to Jorge Galindo.

18 recordsLinked to original sources

arXiv2024

$\ell^1$-bases, algebraic structure and strong Arens irregularity of Banach algebras in harmonic analysis$^1$

A long standing problem in abstract harmonic analysis concerns the strong Arens irregularity (sAir, for short) of the Fourier algebra $A(G)$ of a locally compact group $G.$ The groups for which $A(G)$ is known to be sAir are all amenable. So far this class includes the abelian groups, the discrete amenable groups, the second countable amenable groups $G$ such that $\overline{[G, G]}$ is not open in $G,$ the groups of the form $\prod_{i=0}^\infty G_i$ where each $G_i$, $i\ge1$, is a non-trivial metrizable compact group and $G_0$ is an amenable second countable locally compact group, the groups of the form $G_0\times G$, where $G$ is a compact group whose local weight $w(G)$ has uncountable cofinality and $G_0$ is any locally compact amenable group with $w(G_0)\le w(G)$, and the compact group $SU(2).$ We were primarily concerned with the groups for which $A(G)$ is sAir. We introduce a new class of $\ell^1$-bases in Banach algebras. These new $\ell^1$-bases enable us, among other results, to unify most of the results related to Arens products proved in the past seventy years since Arens defined his products. This includes the strong Arens irregularity of algebras in harmonic analysis, and in particular almost all the cases mentioned above for the Fourier algebras. In addition, we also show that $A(G)$ is sAir for compact connected groups with an infinite dual rank. The $\ell^1$-bases for the Fourier algebra are constructed with coefficients of certain irreducible representations of the group. With this new approach using $\ell^1$-bases, the rich algebraic structure of the algebras and semigroups under study such as the second dual of Banach algebras with an Arens product or certain semigroup compactifications (the Stone-\v Cech compactification of an infinite discrete group, for instance) is also unveiled

math.FA↗

arXiv2024

Positive definite functions as uniformly ergodic multipliers of the Fourier algebra

Let G be a locally compact group and let $ϕ$ be a positive definite function on G with $ϕ(e)=1$. This function defines a multiplication operator $M_ϕ$ on the Fourier algebra $A(G)$ of $G$. The aim of this paper is to classify the ergodic properties of the operators $M_ϕ$, focusing on several key factors, including the subgroup $H_ϕ=\{x\in G\colon ϕ(x)=1\}$, the spectrum of $M_ϕ$, or how ``spread-out'' a power of $M_ϕ$ can be. We show that the multiplication operator $M_ϕ$ is uniformly mean ergodic if and only if $H_ϕ$ is open and 1 is not an accumulation point of the spectrum of $M_ϕ$. Equivalently, this happens when some power of $ϕ$ is not far, in the multiplier norm, from a function supported on finitely many cosets of $H_ϕ$. Additionally, we show that the powers of $M_ϕ$ converge in norm if, and only if, the operator is uniformly mean ergodic and $H_ϕ=\{x\in G\colon |ϕ(x)|=1\}$.

math.FA↗

arXiv2024

A note on a question of Garth Dales: Arens regularity as a three space property

Garth Dales asked whether a Banach algebra $\mathcal{A}$ having an Arens regular closed ideal $\mathcal{J}$ with Arens regular quotient $\mathcal{A}/\mathcal{J}$ is necessarily Arens regular. We prove in this note that, for a class of Banach algebras including the standard algebras in harmonic analysis, Garth's conditions force the algebra to be even reflexive. We also give examples of Banach algebras with Garth's conditions, that are not Arens regular.

math.FA↗

arXiv2024

Arens regularity and irregularity of ideals in Fourier and group algebras

Let $\mathcal{A}$ be a weakly sequentially complete Banach algebra containing a bounded approximate identity that is an ideal in its second dual $\mathcal{A}^{\ast\ast}$, we call such an algebra a Wesebai algebra. In the present paper we examine the Arens regularity properties of closed ideals of algebras in the Wesebai class. We observe that, although Wesebai algebras are always strongly Arens irregular, a variety of Arens regularity properties can be observed within their closed ideals. After characterizing Arens regular ideals and strongly Arens irregular ideals, we proceed to particularize to the main examples of \wasabi algebras, the convolution group algebras $L^1(G)$, $G$ compact, and the Fourier algebras $A(\Gamma)$, $\Gamma$ discrete and amenable. We find examples of Arens regular ideals in $L^1(G)$ and $A(\Gamma)$, both reflexive and nonreflexive and examples of strongly Arens irregular ideals that are not in the \wasabi class. For this, we construct, in many noncommutative groups, a new class of Riesz sets which are not $\Lambda(p)$, for any $p>1$. Our approach also shows that every infinite Abelian group contains a Rosenthal set that is not $\Lambda(p)$, for any $p>0$. These latter results could be of independent interest.

math.FA↗

arXiv2023

Uniformly ergodic probability measures

Let $G$ be a locally compact group and $μ$ be a probability measure on $G$. We consider the convolution operator $λ_1(μ)\colon L_1(G)\to L_1(G)$ given by $λ_1(μ)f=μ\ast f$ and its restriction $λ_1^0(μ)$ to the augmentation ideal $L_1^0(G)$. Say that $μ$ is uniformly ergodic if the Cesàro means of the operator $λ_1^0(μ)$ converge uniformly to 0, that is, if $λ_1^0(μ)$ is a uniformly mean ergodic operator with limit 0 and that $μ$ is uniformly completely mixing if the powers of the operator $λ_1^0(μ)$ converge uniformly to 0. We completely characterize the uniform mean ergodicity of the operator $λ_1(μ)$ and the uniform convergence of its powers and see that there is no difference between $λ_1(μ)$ and $λ_1^0(μ) $ in this regard. We prove in particular that $μ$ is uniformly ergodic if and only if $G$ is compact, $μ$ is adapted (its support is not contained in a proper closed subgroup of $G$) and 1 is an isolated point of the spectrum of $μ$. The last of these three conditions is actually equivalent to $μ$ being spread-out (some convolution power of $μ$ is not singular). The measure $μ$ is uniformly completely mixing if and only if $G$ is compact, $μ$ is spread-out and the only unimodular value of the spectrum of $μ$ is 1.

math.FA↗

arXiv2020

Orthogonal $\ell_1$-sets and extreme non-Arens regularity of preduals of von Neumann algebras

We propose a new definition for a Banach algebra $\mathfrak{A}$ to be extremely non-Arens regular, namely that the quotient $\mathfrak{A}^\ast/\mathscr{WAP}(\mathfrak{A})$ of $\mathfrak{A}^\ast$ with the space of its weakly almost periodic elements contains an isomorphic copy of $\mathfrak{A}^\ast.$ This definition is simpler and formally stronger than the original one introduced by Granirer in the nineties. We then identify sufficient conditions for the predual $\mathfrak{V}_\ast$ of a von Neumann algebra $\mathfrak{V}$ to be extremely non-Arens regular in this new sense. These conditions are obtained with the help of orthogonal $\ell_1$-sets of $\mathfrak{V}_\ast.$ We show that some of the main algebras in Harmonic Analysis satisfy these conditions. Among them,there is ${\small \bullet}$ the weighted semigroup algebra of any weakly cancellative discrete semigroup, for any diagonally bounded weight, ${\small \bullet}$ the weighted group algebra of any non-discrete locally compact infinite group and for any weight, ${\small \bullet}$ the weighted measure algebra of any locally compact infinite group, for any diagonally bounded weight, ${\small \bullet}$ the Fourier algebra of any locally compact infinite group having its local weight greater or equal than its compact covering number, ${\small \bullet}$ the Fourier algebra of any countable discrete group containing an infinite amenable subgroup.

math.FA↗

arXiv2020

On the extreme non-Arens regularity of Banach algebras

As is well-know, on an Arens regular Banach algebra all continuous functionals are weakly almost periodic. In this paper we show that $\ell^1$-bases which approximate upper and lower triangles of products of elements in the algebra produce large sets of functionals that are not weakly almost periodic. This leads to criteria for extreme non-Arens regularity of Banach algebras in the sense of Granirer. We find in particular that bounded approximate identities (bai's) and bounded nets converging to invariance (TI-nets) both fall into this approach, suggesting that this is indeed the main tool behind most known constructions of non-Arens regular algebras. These criteria can be applied to the main algebras in harmonic analysis such as the group algebra, the measure algebra, the semigroup algebra (with certain weights) and the Fourier algebra. In this paper, we apply our criteria to the Lebesgue-Fourier algebra, the 1-Segal Fourier algebra and the Figà-Talamanca Herz algebra.

math.FA↗

arXiv2020

Ergodic properties of convolution operators

Let $G$ be a locally compact group and $μ$ be a measure on $G$. In this paper we find conditions for the convolution operators $λ_p(μ)$, defined on $L^p(G)$ and given by convolution by $μ$, to be mean ergodic and uniformly mean ergodic. The ergodic properties of the operators $λ_p(μ)$ are related to the ergodic properties of the measure $μ$ as well.

math.FA↗

arXiv2018

Sampling Almost Periodic and related Functions

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, what makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps.

math.FA↗

arXiv2017

Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points

The spectrum of an admissible subalgebra $\mathscr{A}(G)$ of $\mathscr{LUC}(G)$, the algebra of right uniformly continuous functions on a locally compact group $G$, constitutes a semigroup compactification $G^\mathscr{A}$ of $G$. In this paper we analyze the algebraic behaviour of those points of $G^\mathscr{A}$ that lie in the closure of $\mathscr{A}(G)$-sets, sets whose characteristic function can be approximated by functions in $\mathscr{A}(G)$. This analysis provides a common ground for far reaching generalizations of Veech's property (the action of $G$ on $G^\mathscr{LUC}$ is free) and Pym's Local Structure Theorem. This approach is linked to the concept of translation-compact set, recently developed by the authors, and leads to characterizations of strongly prime points in $G^\mathscr{A}$, points that do not belong to the closure of $G^\ast G^\ast$, where $G^\ast=G^\mathscr{A}\setminus G.$ All these results will be applied to show that, in many of the most important algebras, left invariant means of $\mathscr{A}(G)$ (when such means are present) are supported in the closure of $G^\ast G^\ast$.

math.FA↗

arXiv2013

Interpolation sets and the size of quotients of function spaces on a locally compact group

We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation sets and unifies the approaches followed by many authors to obtain particular cases. Among the applications we find, we obtain that the quotients WAP(G)/B(G) (G being a locally compact group in the class [IN] or a nilpotent locally compact group) and CB(G)/LUC(G) (G being any non-compact non-discrete locally compact group) contain a linearly isometric copy of \ell_\infty(κ(G)) where κ(G) is the compact covering number of G, and WAP(G), B(G) and LUC(G) refer, respectively, to the algebra of weakly almost periodic functions, the uniform closure of the Fourier-Stieltjes algebra and the bounded right uniformly continuous functions.

math.FA↗

arXiv2013

Extreme non-Arens regularity of the group algebra

Following Granirer, a Banach algebra A is extremely non-Arens regular when the quotient space A*/WAP(A) contains a closed linear subspace which has A* as a continuous linear image. We prove that the group algebra L^1(G) of any infinite locally compact group is always extremely non-Arens regular. When G is not discrete, this result is deduced from the much stronger property that, in fact, there is a linear isometric copy of L^\infty(G) in the quotient space L^\infty(G)/CB(G), where CB(G) stands for the algebra of all continuous and bounded functions on G.

math.FA↗

arXiv2012

Reflexivity in precompact groups and extensions

We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that: (1) A precompact Abelian group G of bounded order is reflexive iff the dual group $\hat{G}$ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G. (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive $ω$-bounded group (even dual to a reflexive P-group) can fail to be reflexive. We also show that the P-modification of a reflexive $σ$-compact group can be nonreflexive (even if the P-modification of a locally compact Abelian group is always reflexive).

math.GN↗

arXiv2011

Approximable WAP- and LUC-interpolation sets

Extending and unifying concepts extensively used in the literature, we introduce the notion of approximable interpolation sets for algebras of functions on locally compact groups, especially for weakly almost periodic functions and for uniformly continuous functions. We characterize approximable interpolation sets both in combinatorial terms and in terms of the $\mathscr{LUC}$- and $\mathscr{WAP}$-compactifications and analyze some of their properties.

math.GN↗

arXiv2010

Pseudocompact group topologies with no infinite compact subsets

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property $\h$). Every pseudocompact Abelian group $G$ with cardinality $|G|\leq 2^{2^\cc}$ satisfies this inequality and therefore admits a pseudocompact group topology with property $\h$. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property $\h$. We also observe that pseudocompact Abelian groups with property $\h$ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.

math.GR↗

arXiv2010

Nondiscrete P-Groups Can be Reflexive

We present a series of examples of nondiscrete reflexive P-groups (i.e., groups in which all $G_δ$-sets are open) as well as noncompact reflexive $ω$-bounded groups (in which the closure of every countable set is compact). Our main result implies that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is a reflexive group. In particular, every compact Abelian group with the P-modified topology is reflexive. This answers a question posed by S. Hernández and P. Nickolas and solves a problem raised by Ardanza-Trevijano, Chasco, Dom\'ınguez, and Tkachenko.

math.GN↗

arXiv2007

Characterizing group $C^\ast$-algebras through their unitary groups: the Abelian case

We study to what extent group $C^\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^\ast(Γ)$, such as the $K$-theoretic $K_1(C^\ast(Γ))$ and find that $C^\ast$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^\ast(Γ)$ (even $Γ$) is characterized by the topological group structure of its unitary group when $Γ$ is torsion-free and Abelian.

math.OA↗

arXiv2006

On unitary representability of topological groups

We prove that the additive group $(E^\ast,τ_k(E))$ of an $\mathscr{L}_\infty$-Banach space $E$, with the topology $τ_k(E)$ of uniform convergence on compact subsets of $E$, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is \emph{unitarily representable}). This is the same as proving that the topological group $(E^\ast,τ_k(E))$ is uniformly homeomorphic to a subset of $\ell_2^κ$ for some $κ$. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative $C^\ast$-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

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