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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.
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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 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.
We give the beginnings of the development of a theory of what we call R-coactions of a locally compact group on a $C^*$-algebra. These are the coactions taking values in the maximal tensor product, as originally proposed by Raeburn. We show that the theory has some gaps as compared to the more familiar theory of standard coactions. However, we indicate how we needed to develop some of the basic properties of R-coactions as a tool in our program involving the use of coaction functors in the study of the Baum-Connes conjecture.
We define a strong Morita-type equivalence $sim _{sigma Delta }$ for operator algebras. We prove that $Asim _{sigma Delta }B$ if and only if $A$ and $B$ are stably isomorphic. We also define a relation $subset _{sigma Delta }$ for operator algebras. We prove that if $A$ and $B$ are $C^*$-algebras, then $Asubset _{sigma Delta } B$ if and only if there exists an onto $*$-homomorphism $theta :Botimes mathcal K rightarrow Aotimes mathcal K,$ where $mathcal K$ is the set of compact operators acting on an infinite dimensional separable Hilbert space. Furthermore, we prove that if $A$ and $B$ are $C^*$-algebras such that $Asubset _{sigma Delta } B$ and $Bsubset _{sigma Delta } A $, then there exist projections $r, hat r$ in the centers of $A^{**}$ and $B^{**}$, respectively, such that $Arsim _{sigma Delta }Bhat r$ and $A (id_{A^{**}}-r) sim _{sigma Delta }B(id_{B^{**}}-hat r). $
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