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Register a new userCo-universal C*-algebras associated to generalised graphs
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We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_{min}(Lambda) which is co-universal for partial-isometric representations of Lambda which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent C*_{min}(Lambda) for some (N^2 * N)-graph Lambda.
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To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C*-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.
A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnems strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.
Let $G$ be a Hausdorff, etale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
We extend the usual theory of universal C*-algebras from generators and relations in order to allow some relations to be described using the strong operator topology. In particular, we can allow some infinite sum relations. We prove a universal property for the algebras we define and we show how the Cuntz algebra of infinite isometries as well as the Exel-Laca algebras can be described using infinite sum relations. Finally, we give some sufficient conditions for when a C*-algebra generated by projections and partial isometries is a universal C*-algebra using only norm relations, in case one still wants to avoid using relations with respect to the strong operator topology.
In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $Lambda$. We begin by giving an alternative characterization of the $Lambda$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.
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