Exel–Pardo algebras with a twist

Abstract

Takeshi Katsura associated a $C^{\ast }$-algebra $C_{A,B}^{\ast }$ to a pair of square matrices $A\ge 0$ and $B$ of the same size with integral coefficients and gave sufficient conditions on $(A,B)$ to be simple purely infinite (SPI). We call such a pair a KSPI pair. It follows from a result of Katsura that any separable $C^{\ast }$-algebra $\mathfrak{A}$ which is a cone of a map $\xi :C(S^1{)}^n\to C(S^1{)}^n$ in Kasparov’s triangulated category $KK$ is $KK$-isomorphic to $C_{A,B}^{\ast }$ for some KSPI pair $(A,B)$. In this article, we introduce, for the data of a commutative ring $\ell$, non-necessarily square matrices $A$, $B$ and a matrix $C$ of the same size with coefficients in the group $\mathcal{U}(\ell )$ of invertible elements, an $\ell$-algebra ${\mathcal{O}}_{A,B}^C$, the twisted Katsura algebra of the triple $(A,B,C)$. When $A$ and $B$ are square and $C$ is trivial, we recover the Katsura $\ell$-algebra first considered by Enrique Pardo and Ruy Exel. We show that if $\ell$ is a field of characteristic $0$ and $(A,B)$ is KSPI, then ${\mathcal{O}}_{A,B}^C$ is SPI, and that any $\ell$-algebra which is a cone of a map $\xi :\ell [t,t^{-1}{]}^n\to \ell [t,t^{-1}{]}^n$ in the triangulated bivariant algebraic $K$-theory category $kk$ is $kk$-isomorphic to ${\mathcal{O}}_{A,B}^C$ for some $(A,B,C)$ as above so that $(A,B)$ is KSPI. Katsura $\ell$-algebras are examples of the Exel–Pardo algebras $L(G,E,\varphi )$ associated to a group $G$ acting on a directed graph $E$ and a $1$-cocycle $\varphi :G\times E^1\to G$. Similarly, twisted Katsura algebras are examples of the twisted Exel–Pardo $\ell$-algebras $L(G,E,{\varphi }_c)$ we introduce in the current article; they are associated to data $(G,E,\varphi )$ as above twisted by a $1$-cocycle $c:G\times E^1\to \mathcal{U}(\ell )$. The algebra $L(G,E,{\varphi }_c)$ can be variously described by generators and relations, as a quotient of a twisted semigroup algebra, as a twisted Steinberg algebra, as a corner skew Laurent polynomial algebra, and as a universal localization of a tensor algebra. We use each of these guises of $L(G,E,{\varphi }_c)$ to study its $K$-theoretic, regularity, and (purely infinite) simplicity properties. For example, we show that if $\ell$ is a field of characteristic $0$, $G$ and $E$ are countable, and $E$ is regular, then $L(G,E,{\varphi }_c)$ is simple whenever the Exel–Pardo $C^{\ast }$-algebra $C^{\ast }(G,E,\varphi )$ is, and is SPI if in addition the Leavitt path algebra $L(E)$ is SPI.