arXiv subjects

Krzysztof Bardadyn

Publications and source records attributed to Krzysztof Bardadyn.

3 recordsLinked to original sources

arXiv2026

Banach algebras associated to twisted étale groupoids: simplicity and pure infiniteness

We define reduced and essential Banach algebras associated to a twisted étale (not necessarily Hausdorff) groupoid $(\mathcal{G},\mathcal{L})$ and extend some fundamental results from $C^*$-algebras to this context. We prove that for topologically free groupoids the associated essential Banach algebras have the ideal interesection property, and thus such an algebra is simple if and only if the groupoid is minimal. We give conditions under which reduced algebras are essential (for example Hausdorffness of $\mathcal{G}$ is sufficient). This in particular solves the simplicity problem posed recently by Gardella-Lupini for $L^p$-operator algebras associated to $\mathcal{G}$. In addition, using either the $n$-filling or locally contracting condition we give pure infiniteness criteria for essential simple Banach algebras associated to $(\mathcal{G},\mathcal{L})$. This extends the corresponding $C^*$-algebraic results that were previously known to hold in the untwisted Hausdorff case. The results work nicely, and allow for characterisation of the generalized intersection property, in the realm of $L^P$-operator algebras where $P \subseteq [1,\infty]$ is a non-empty set of parameters. Such algebras cover in particular $L^p$-operator algebras, for $p\in [1,\infty]$, and their Banach $*$-algebra versions. We apply our results to Banach algebra crossed products by twisted partial group actions, Roe-type Banach algebras with coefficients in finite-rank operators on a Banach space, twisted tight $L^P$-operator algebras of inverse semigroups, graph $L^P$-operator algebras, and algebras associated to self-similar group actions on graphs. We also interpret our results in terms of twisted inverse semigroup actions and their crossed products.

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arXiv2025

Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces

We introduce Banach algebras associated to twisted étale groupoids $(\mathcal{G},\mathcal{L})$ and to twisted inverse semigroup actions. This provides a unifying framework for numerous recent papers on $L^p$-operator algebras and the theory of groupoid $C^*$-algebras. We prove disintegrations theorems that allow to study Banach algebras associated to $(\mathcal{G},\mathcal{L})$ as universal Banach algebras generated by $C_0(X)$ and a twisted inverse semigroup $S$ of partial isometries subject to some relations. They work best when the target of a representation is a dual Banach algebra. For representations on dual Banach spaces, they allow to extend representations to twisted Borel convolution algebras, which is crucial when the groupoid is non-Hausdorff. We establish fundamental norm estimates and hierarchy for full and reduced $L^p$-operator algebras for $(\mathcal{G},\mathcal{L})$ and $p \in [1,\infty]$, whose special cases have been studied recently by Gardella-Lupini, Choi-Gardella-Thiel and Hetland-Ortega. We show that in the constructions of $L^p$-analogues of Cuntz or graph algebras, by Phillips and Cortiñas-Rodr\'ıguez, the use of spatial partial isometries is not an assumption, in fact it is forced by the relations. We also introduce tight inverse semigroup Banach algebras that cover ample groupoid Banach algebras, and discuss Banach algebras associated to directed graphs. Our results cover non-Hausdorff étale groupoids and both real and complex algebras. Some of the results are new already for complex $C^*$-algebras.

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arXiv2023

$L^p$-Cuntz algebras and spectrum of weighted composition operators

Let $p\in [1,\infty)$. We define an $L^p$-operator algebra crossed product by a transfer operator for the topological Bernoulli shift $φ$ on $X=\{1,...,n\}^{\mathbb{N}}$, and we prove it is isometrically isomorphic to the $L^p$-analog $\mathcal{O}_n^p$ of the Cuntz algebra introduced by Phillips. As an application, we prove that the spectrum of the associated `abstract weighted shift operators' $aT$, $a\in C(X)$, is a disk with a radius given by the formula $$ r(aT)=\max_{μ\in\text{Erg}(X,φ)} \exp\left( \int_X \ln(|a|\varrho^{\frac{1}{p}})dμ+ \frac{h_φ(μ)}{p} \right) $$ where $\varrho$ is a potential associated to the transfer operator, and $h_φ(μ)$ is Kolmogorov-Sinai entropy. This generalizes classical results for $p=2$.

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