Realising the C*-algebra of a higher-rank graph as an Exel crossed product

We use the boundary-path space of a finitely-aligned k-graph \Lambda to construct a compactly-aligned product system X, and we show that the graph algebra C^*(\Lambda) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.Comment: Corrections made to Section 5.

Two families of Exel-Larsen crossed products

Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $\Gamma$, and the induced action on $C(\Gamma)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $\Gamma$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ on the UHF cores in 2-graph algebras, as introduced by Yang, and confirm that these actions have properties similar to those of single endomorphisms of the core in Cuntz algebras.Comment: 17 page

Leavitt RR-algebras over countable graphs embed into L2,RL_{2,R}

For a commutative ring $R$ with unit we show that the Leavitt path algebra $L_R(E)$ of a graph $E$ embeds into $L_{2,R}$ precisely when $E$ is countable. Before proving this result we prove a generalised Cuntz-Krieger Uniqueness Theorem for Leavitt path algebras over $R$.Comment: 17 pages. At the request of a referee the previous version of this paper has been split into two papers. This version is the first of these papers. The second will also be uploaded to the arXi

C*-Algebras of algebraic dynamical systems and right LCM semigroups

We introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group. To each algebraic dynamical system we associate a C*-algebra and describe it as a semigroup C*-algebra. As part of our analysis of these C*-algebras we prove results for right LCM semigroups. More precisely we discuss functoriality of the full semigroup C*-algebra and compute its K-theory for a large class of semigroups. We introduce the notion of a Nica-Toeplitz algebra of a product system over a right LCM semigroup, and show that it provides a useful alternative to study algebraic dynamical systems.Comment: 28 pages, to appear in Indiana Univ. Math.

On C*-algebras associated to right LCM semigroups

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.Comment: 31 page

L2,Z⊗L2,ZL_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}} does not embed in L2,ZL_{2,\mathbb{Z}}

For a commutative ring $R$ with unit we investigate the embedding of tensor product algebras into the Leavitt algebra $L_{2,R}$. We show that the tensor product $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$ (as a unital $*$-algebra). We also prove a partial non-embedding result for the more general $L_{2,R} \otimes L_{2,R}$. Our techniques rely on realising Thompson's group $V$ as a subgroup of the unitary group of $L_{2,R}$.Comment: 16 pages. At the request of a referee the paper arXiv:1503.08705v2 was split into two papers. This is the second of those paper

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their C∗C∗-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs EE we construct a groupoid G(C∗(E),D(E))G(C∗(E),D(E)) from the graph algebra C∗(E)C∗(E) and its diagonal subalgebra D(E)D(E) which generalises Renault’s Weyl groupoid construction applied to (C∗(E),D(E))(C∗(E),D(E)). We show that G(C∗(E),D(E))G(C∗(E),D(E)) recovers the graph groupoid GEGE without the assumption that every cycle in EE has an exit, which is required to apply Renault’s results to (C∗(E),D(E))(C∗(E),D(E)). We finish with applications of our results to out-splittings of graphs and to amplified graphs

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto's notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^*$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ from the graph algebra $C^*(E)$ and its diagonal subalgebra $\mathcal{D}(E)$ which generalises Renault's Weyl groupoid construction applied to $(C^*(E),\mathcal{D}(E))$. We show that $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ recovers the graph groupoid $\mathcal{G}_E$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault's results to $(C^*(E),\mathcal{D}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.Comment: 27 page