arXiv subjects

Bartosz K. Kwaśniewski

Publications and source records attributed to Bartosz K. Kwaśniewski.

4 recordsLinked to original sources

arXiv2026

Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras

We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff étale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable finiteness of the reduced and essential C*-algebras. If the reduced C*-algebra of a twisted groupoid is simple and the type semigroup satisfies a weak version of almost unperforation, then the C*-algebra is either stably finite or purely infinite. We apply our theory to Cartan inclusions. We calculate the type semigroup for the possibly non-Hausdorff groupoids associated to self-similar group actions on graphs and deduce a dichotomy for the resulting Exel-Pardo algebras.

math.OA↗

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.

math.FA↗

arXiv2020

Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles

We generalise various non-triviality conditions for group actions to Fell bundles over discrete groups and prove several implications between them. We also study sufficient criteria for the reduced section C*-algebra C_r(B) of a Fell bundle (B_g) to be strongly purely infinite. If the unit fibre A=B_e contains an essential ideal that is separable or of Type I, then the Fell bundle is aperiodic if and only if it is topologically free. If, in addition, G=Z or G=Z/p for a square-free number p, then these equivalent conditions are satisfied if and only if A detects ideals in C_r(B), if and only if A^+ \ {0} supports C_r(B) in the Cuntz sense. For G as above and arbitrary A, C_r(B) is simple if and only if the Fell bundle B is minimal and pointwise outer. In general, B is aperiodic if and only if each of its non-trivial fibres has a non-trivial Connes spectrum. If G is finite or if A contains an essential ideal that is of Type I or simple, then aperiodicity is equivalent to pointwise pure outerness.

math.OA↗

arXiv2018

Nica-Toeplitz algebras associated with right tensor $C^*$-precategories over right LCM semigroups

We introduce and analyze the full $\mathcal{NT}_{\mathcal{L}}(\mathcal{K})$ and the reduced $\mathcal{NT}_{\mathcal{L}}^r(\mathcal{K})$ Nica-Toeplitz algebra associated to an ideal $\mathcal{K}$ in a right tensor $C^*$-precategory $\mathcal{L}$ over a right LCM semigroup $P$. Our main results are uniqueness theorems in the spirit of classical Coburn's theorem, generalizing uniqueness results for Toeplitz-type $C^*$-algebras associated to single $C^*$-correspondences, quasi-lattice ordered semigroups, and crossed products twisted by product systems of $C^*$-correspondences obtained by Fowler, Laca and Raeburn. We formulate geometric conditions on a representation $Φ$ of $\mathcal{K}$ so that the $C^*$-algebra it generates, $C^*(Φ(\mathcal{K}))$, naturally lies between $\mathcal{NT}_{\mathcal{L}}^r(\mathcal{K})$ and $\mathcal{NT}_{\mathcal{L}}(\mathcal{K})$. Under suitable amenability hypotheses, $C^*(Φ(\mathcal{K}))$ and $\mathcal{NT}_{\mathcal{L}}(\mathcal{K})$ are isomorphic. The geometric conditions are necessary for our uniqueness result when the right tensoring preserves $\mathcal{K}$ and in general they capture uniqueness of the $C^*$-algebra generated by a natural extension of $Φ$ to $\mathcal{L}$. In particular, the latter algebra could be viewed as a Doplicher-Roberts version of $\mathcal{NT}_{\mathcal{L}}(\mathcal{K})$.

math.OA↗