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We study the external and internal Zappa-Szep product of topological groupoids. We show that under natural continuity assumptions the Zappa-Szep product groupoid is etale if and only if the individual groupoids are etale. In our main result we show that the C*-algebra of a locally compact Hausdorff etale Zappa-Szep product groupoid is a C*-blend, in the sense of Exel, of the individual groupoid C*-algebras. We finish with some examples, including groupoids built from *-commuting endomorphisms, and skew product groupoids.
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We consider Toeplitz and Cuntz-Krieger $C^*$-algebras associated with finitely aligned left cancellative small categories. We pay special attention to the case where such a category arises as the Zappa-Szep product of a category and a group linked by a one-cocycle. As our main application, we obtain a new approach to Exel-Pardo algebras in the case of row-finite graphs. We also present some other ways of constructing $C^*$-algebras from left cancellative small categories and discuss their relationship.
We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{ipi/ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C^*-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras {mathfrak g} eq E_8 and certain roots of unity. Our main tools here are Drinfelds coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfelds associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case {mathfrak g}={mathfrak sl}_N we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.
Let $G$ be a Hausdorff, etale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
Given a normal subgroup bundle $mathcal A$ of the isotropy bundle of a groupoid $Sigma$, we obtain a twisted action of the quotient groupoid $Sigma/mathcal A$ on the bundle of group $C^*$-algebras determined by $mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Sigma)$. Restricting to the case where $mathcal A$ is abelian, we describe $C^*(Sigma)$ as the $C^*$-algebra associated to a $mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Sigma/mathcal A$ on the Pontryagin dual space of $mathcal A$. We give some illustrative examples of this result.
In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup ${rm Cu}(C_r^*(G))$ is almost divisible and ${rm Cu}(C_r^*(G))$ and ${rm Cu}(C_r^*(G)otimes mathcal{Z})$ are canonically order-isomorphic, where $mathcal{Z}$ denotes the Jiang-Su algebra.
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