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Jared T. White

Publications and source records attributed to Jared T. White.

10 recordsLinked to original sources
arXiv2026

Non-Wiener groups with a Gelfand pair

Let $G$ be a non-amenable locally compact group and $K$ a compact subgroup of $G$ such that $(G,K)$ is a Gelfand pair. We show that if $G$ admits a suitable boundary representation which is topologically irreducible and not unitarizable, then $G$ is not a Wiener group in the sense that its Fourier transform does not satisfy the analogue of Wiener's Tauberian theorem. As an application, we show that if $G$ is a closed non-compact boundary transitive group of automorphisms of a connected locally finite graph with infinitely many ends, or a non-abelian split reductive algebraic group over a non-archimedean local field, then $G$ is not Wiener.

math.FA↗
arXiv2026

Radicals of Biduals of Beurling Algebras Can Be Different for the Two Arens Products

Let $\operatorname{rad}$ denote the Jacobson radical of a Banach algebra, and let $\Box$ and $\Diamond$ denote the two Arens products on its bidual. We give an example of a Beurling algebra $\mathcal{A}$ for which $\operatorname{rad}(\mathcal{A}^{**}, \Box) \neq \operatorname{rad}(\mathcal{A}^{**}, \Diamond)$, answering a question of Dales and Lau. The underlying group in our example is the free group on three generators.

math.FA↗
arXiv2024

Weak*-Simplicity of Convolution Algebras on Discrete Groups

We prove that, given a discrete group $G$, and $1 \leq p < \infty$, the algebra of $p$-convolution operators $CV_p(G)$ is weak*-simple, in the sense of having no non-trivial weak*-closed ideals, if and only if $G$ is an ICC group. This generalises the basic fact that $vN(G)$ is a factor if and only if $G$ is ICC. When $p=1$, $CV_p(G) = \ell^1(G)$. In this case we give a more detailed analysis of the weak*-closed ideals, showing that they can be described in terms of the weak*-closed ideals of $\ell^1(FC(G))$; when $FC(G)$ is finite, this leads to a classification of the weak*-closed ideals of $\ell^1(G)$.

math.FA↗
arXiv2023

On the Dales-Zelazko conjecture for Beurling algebras on discrete groups

Let $G$ be a group which is either virtually soluble or virtually free, and let $ω$ be a weight on $G$. We prove that, if $G$ is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, ω)$ which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Zelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance we describe a Beurling algebra on an infinite group in which every left ideal of finite codimension is finitely generated, and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Zelazko's conjecture.

math.FA↗
arXiv2022

The ideal structure of measure algebras and asymptotic properties of group representations

We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We also prove a version of this result for certain weighted measure algebras. We apply our classification to obtain an analogue of Barnes' Theorem on integrable representations for representations vanishing at infinity. We next study the relationship between weak*-closedness and finite generation, proving that in many cases $M(G)$ has no finitely-generated maximal left ideals. We also show that the measure algebra of the 2D Euclidean rigid motion group has a weak*-closed maximal left ideal that is not generated by a projection, and investigate whether or not it has any weak*-closed left ideals which are not finitely-generated.

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arXiv2021

On algebras associated with invariant means on the subnormal subgroups of an amenable group

Let $G$ be an amenable group. We define and study an algebra $\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $\mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely-generated counterexamples to a question of Dales and Lau, first resolved by the author in a previous article, which asks whether we always have $(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0 \}$. We further study this question by showing that $(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0 \}$ imposes certain structural constraints on the group $G$.

math.GR↗
arXiv2019

Left ideals of Banach algebras and dual Banach algebras

We investigate topologically left Noetherian Banach algebras. We show that if $G$ is a compact group, then $L^{\, 1}(G)$ is topologically left Noetherian if and only if $G$ is metrisable. We prove that, given a Banach space $E$ such that $E'$ has BAP, the algebra of compact operators $\mathcal{K}(E)$ is topologically left Noetherian if and only if $E'$ is separable; it is topologically right Noetherian if and only if $E$ is separable. We then give some examples of dual Banach algebras which are topologically left Noetherian in the weak*-topology. Finally we give a unified approach to classifying the weak*-closed left ideals of certain dual Banach algebras that are also multiplier algebras, with applications to $M(G)$ for $G$ a compact group, and $\mathcal{B}(E)$ for $E$ a reflexive Banach space with AP.

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arXiv2018

Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Given a Banach space $E$, we ask which closed subspaces may be realised as the kernel of a bounded operator $E \rightarrow E$. We prove some positive results which imply in particular that when $E$ is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space $E$ which contains a closed subspace that cannot be realized as the kernel of any bounded operator on $E$. This implies that the Banach algebra $\mathcal{B}(E)$ of bounded operators on $E$ fails to be weak*-topologically left Noetherian. The Banach space $E$ that we use is the dual of Wark's non-separable, reflexive Banach space with few operators.

math.FA↗
arXiv2018

The radical of the bidual of a Beurling algebra

We prove that the bidual of a Beurling algebra on $\mathbb{Z}$, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that ${\rm rad\,}(\ell^{\, 1}(\oplus_{i=1}^\infty \mathbb{Z})")$ contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight $ω$ on $\mathbb{Z}$ such that the bidual of $\ell^{\, 1}(\mathbb{Z}, ω)$ contains a radical element which is not nilpotent.

math.FA↗