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Register a new userOn finitely aligned left cancellative small categories, Zappa-Szep products and Exel-Pardo algebras
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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.
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Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSCs, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.
We generalize a recent construction of Exel and Pardo, from discrete groups acting on finite directed graphs to locally compact groups acting on topological graphs. To each cocycle for such an action, we construct a $C^*$-correspondence whose associated Cuntz-Pimsner algebra is the analog of the Exel-Pardo $C^*$-algebra.
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.
We introduce an algebraic version of the Katsura $C^*$-algebra of a pair $A,B$ of integer matrices and an algebraic version of the Exel-Pardo $C^*$-algebra of a self-similar action on a graph. We prove a Graded Uniqueness Theorem for such algebras and construct a homomorphism of the latter into a Steinberg algebra that, under mild conditions, is an isomorphism. Working with Steinberg algebras over non-Hausdorff groupoids we prove that in the unital case, our algebraic version of Katsura $C^*$-algebras are all isomorphic to Steinberg algebras.
We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $|uv-vu|<2$ such that $uvu^*v^*$ commutes with $u$ and $v,$ then the SCA, $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of the rotation algebra $A_theta$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $thetain (-1/2, 1/2)$ be a rational number. We prove the following: For any $ep>0,$ there exists $dt>0$ satisfying the following: if $u$ and $v$ are two unitary matrices such that $$ |uv-e^{2pi itheta}vu|<dtandeqn {1over{2pi i}}tau(log(uvu^*v^*))=theta, $$ then there exists a pair of unitary matrices $tilde{u}$ and $tilde{v}$ such that $$ tilde{u}tilde{v}=e^{2pi itheta} tilde{v}tilde{u},,, |u-tilde{u}|<epandeqn |v-tilde{v}|<ep. $$ Furthermore, a generalization of this for all real $theta$ is obtained for unitaries in unital infinite dimensional simple $C^*$-algebras of tracial rank zero.
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