Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally-graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit. This is a joint work with Joan Bosa, Roozbeh Hazrat and Aidan Sims.Universidad de Málaga. Campus de Excelencia internacional Andalucía Tec

Tensor products of Leavitt path algebras

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_\infty$ and $L_\infty\otimes L_\infty$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\otimes L_2)=0$ and $K_*(L_\infty)=K_*(L_\infty\otimes L_\infty)=K_*(k)$.Comment: 10 pages. Added hypothesis to Corolary 4.5; Example 5.2 expanded, other cosmetic changes, including an e-mail address and some dashes. Final version, to appear in PAM

Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras

We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algebras, $\mathcal{O}(E,C)$ and $\mathcal{O}^r(E,C)$, of a separated graph $(E,C)$, and of the abelianized Leavitt path algebra $L_K^{\text{ab}}(E,C)$ as well. These algebras are the (reduced) crossed products with respect to the above-mentioned partial actions, and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary $D^{\infty}$-saturated subsets of a certain infinite separated graph $(F_{\infty},D^{\infty})$ built from $(E,C)$, called the separated Bratteli diagram of $(E,C)$. We finally use these tools to study simplicity and primeness of the tame separated graph algebras.Comment: 60 page

Maximal C*-algebras of quotients and injective envelopes of C*-algebras

A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.Comment: 37 page

A not so simple local multiplier algebra

We construct an AF-algebra $A$ such that its local multiplier algebra $M_{\text{loc}}(A)$ does not agree with $M_{\text{loc}}(M_{\text{loc}}(A))$, thus answering a question raised by G.K. Pedersen in 1978.Comment: 18 page

Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions

We attach to each finite bipartite separated graph (E,C) a partial dynamical system (\Omega(E,C), F, \theta), where \Omega(E,C) is a zero-dimensional metrizable compact space, F is a finitely generated free group, and {\theta} is a continuous partial action of F on \Omega(E,C). The full crossed product C*-algebra O(E,C) = C(\Omega(E,C)) \rtimes_{\theta} F is shown to be a canonical quotient of the graph C*-algebra C^*(E,C) of the separated graph (E,C). Similarly, we prove that, for any *-field K, the algebraic crossed product L^{ab}_K(E,C) = C_K(\Omega(E,C)) \rtimes_\theta^{alg} F is a canonical quotient of the Leavitt path algebra L_K(E,C) of (E,C). The monoid V(L^{ab}_K(E,C)) of isomorphism classes of finitely generated projective modules over L^{ab}_K(E,C) is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs. Using this, we are able to construct an action of a finitely generated free group F on a zero-dimensional metrizable compact space Z such that the type semigroup S(Z, F, K) is not almost unperforated, where K denotes the algebra of clopen subsets of Z. Finally we obtain a characterization of the separated graphs (E,C) such that the canonical partial action of F on \Omega(E,C) is topologically free.Comment: Final version to appear in Advances in Mathematic

The regular algebra of a quiver

Let $K$ be a fixed field. We attach to each column-finite quiver $E$ a von Neumann regular $K$-algebra $Q(E)$ in a functorial way. The algebra $Q(E)$ is a universal localization of the usual path algebra $P(E)$ associated with $E$. The monoid of isomorphism classes of finitely generated projective right $Q(E)$-modules is explicitly computed.Comment: 29 page