Algebraic Cuntz-Pimsner rings

From a system consisting of a right non-degenerate ring $R$, a pair of $R$-bimodules $Q$ and $P$ and an $R$-bimodule homomorphism $\psi:P\otimes Q\longrightarrow R$ we construct a $\Z$-graded ring $\mathcal{T}_{(P,Q,\psi)}$ called the Toeplitz ring and (for certain systems) a $\Z$-graded quotient $\mathcal{O}_{(P,Q,\psi)}$ of $\mathcal{T}_{(P,Q,\psi)}$ called the Cuntz-Pimsner ring. These rings are the algebraic analogs of the Toeplitz $C^*$-algebra and the Cuntz-Pimsner $C^*$-algebra associated to a $C^*$-correspondence (also called a Hilbert bimodule). This new construction generalizes for example the algebraic crossed product by a single automorphism, corner skew Laurent polynomial ring by a single corner automorphism and Leavitt path algebras. We also describe the structure of the graded ideals of our graded rings in terms of pairs of ideals of the coefficient ring.Comment: 55 pages. Version 3 is a complete rewrite of version 2. In version 4 Def. 3.14, Def. 4.6, Def. 4.8 and Remark 4.9 have been added and some minor adjustments have been mad

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

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic.Comment: 12 pages. Minor corrections. This is the version that will be publishe

Flow Equivalence of G-SFTs

In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata.Comment: The paper has been augmented considerably and the second version is now 81 pages long. This version has been accepted for publication in Transactions of the American Mathematical Societ

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