Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2406.00466

The Zappa-Szép product of twisted groupoids

Anna Duwenig·Boyu Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

We define and study the external and the internal Zappa-Szép product of twists over groupoids. We determine when a pair $(Σ_{1},Σ_{2})$ of twists over a matched pair $(\mathcal{G}_{1},\mathcal{G}_{2})$ of groupoids gives rise to a Zappa-Szép twist $Σ$ over the Zappa-Szép product $\mathcal{G}_{1}\bowtie\mathcal{G}_{2}$. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ is a C*-blend of its subalgebras corresponding to the subtwists $Σ_{i}\to \mathcal{G}_{i}$. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ of two twists $Σ_{1}\to \mathcal{G}_{1}$ and $Σ_{2}\to \mathcal{G}_{2}$.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Anna Duwenig, Boyu Li. 2024-06-01. The Zappa-Szép product of twisted groupoids. https://arxiv.org/abs/2406.00466

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all \( x \in L_p(\mathcal{M}) \), $1<p<\infty$ extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, similar inequalities as in \eqref{abstract1stin} hold for arbitrary $L_\infty-L_1$ contractions with the same condition in the numerical range, yielding a vast generalization of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. Moreover, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Bellow \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalization of Stein's method of embedding maximal function into analytic family of operators. However, we show that even in the classical setting, the variational inequality extending \eqref{abstract1stin} holds for arbitrary operators described above, precisely when the spectrum meets the unit circle only at $1.$

math.OA↗
arXiv2026

Group actions on von Neumann algebras with compact open subgroups

We study strictly outer actions of locally compact groups with a compact open subgroup on von Neumann factors. For amenable groups, we prove 2-cohomology vanishing and obtain classification results using a description of the central sequence algebra and Rohlin-type observations. We also characterize the inclusions of factors associated with group actions, and extend M. Choda's result to this locally compact setting.

math.OA↗
arXiv2026

Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras

We consider an action of an inverse semigroup on a C*-algebra $A$ and use it to construct a groupoid of germs with unit space the spectrum of $A$. Motivated by the representation theory of C*-algebras of groupoids, we construct a concrete family of covariant representations for the action. We use this family to give new definitions of the reduced and essential crossed product C*-algebras that avoid, respectively, passing to the double commutant and local multiplier algebra of $A$. Our reduced crossed product is isomorphic to the one defined by Exel, Buss and Meyer, and when the inverse semigroup is quasi-countable our essential crossed product is isomorphic to the one defined by Kwaśniewski and Meyer.

math.OA↗