Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

Abstract

Abstract. We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra. IntroductionA higher-rank graph is a countable category Λ endowed with a degree functor d : Λ → N k satisfying the unique factorization property: For all λ ∈ Λ and m, n ∈ N k with d(λ) = m + n, there are unique elements µ, ν ∈ Λ such that d(µ) = m, d(ν) = n and λ = µν. The rank of Λ is k and for this reason, (Λ, d) is also called a k-graph. A 1-graph is simply the finite-path category generated freely by an ordinary directed graph.1 In [2], Kumjian and Pask introduced the notion of a higher-rank graph in order to capture the essential features of the C * -algebras that Robertson and Steger associated to buildings [14,15,16,17] and to provide links between these and higher order shift dynamical systems. (See [3] also.)The C * -algebras associated to higher-rank graphs are generalizations of ordinary graph C * -algebras in that they are generated by families of partial isometries {s λ } λ∈Λ that satisfy certain relations that have received a lot of attention in recent years. Kumjian and Pask defined and studied the C * -algebra of (Λ, d), C * (Λ), in terms of a certain type of groupoid that encodes the graph. They were motivated by, and generalized, the theory in [5] which has been the source of considerable inspiration in our subject. However, just as in the setting of ordinary graphs, where the groupoid techniques of [5] require hypotheses that rule out many interesting examples, the work of Kumjian and Pask requires hypotheses that place significant limitations on the nature of the k-graphs that may be analyzed. Our first objective in this paper, then, is to overcome the limitations that Kumjian and Pask place on their k-graphs and to show how to build a groupoid that gives the C * -algebra of an arbitrary k-graph subject only to the condition that it is "finitely aligned" (see Definition 3.4). This condition seems to lie at the natural "boundary" of the subject. That is, with or without the use of groupoids, little can be said about k-graphs that are not finitely aligned.Date: March 10, 2018. 1991 Mathematics Subject Classification. Primary 46L05; Secondary 22A22; 20M18. Key words and phrases. graph algebra; Cuntz-Krieger algebra; higher-rank graph; groupoid; inverse semigroup.The research of the first two authors was supported in part by a grant from the National Science Foundation, DMS-0070405.1 For the purpose of motivation, we discuss the theory of 1-graphs at some length in the next section. For a very readable account of graph C * -algebras, including the rudiments of the C * -algebras of k-graphs, we recommend the CBMS lectures by Iain Raeburn [9]. We were motivated in part by the important contribution of Paterson [8] in which he successfully circumvented the limitations of [5] that involve finiteness hypotheses on the graphs under consideration by first introducing an inverse semi...

Inverse semigroups and combinatorial C*-algebras

“…Although we have not invested all of the necessary energy to study the inverse semigroup constructed from a general higher rank graph, as in [13], we conjecture that the groupoid there denoted by G Λ | ∂Λ is the same as the groupoid G tight of Theorem (13.3) below, or at least our findings seem to give strong indications that this is so. Should this be confirmed, the assertion made in the introduction of [13] that their groupoid is fairly far removed from the universal groupoid of S Λ might need rectification.…”

“…Should this be confirmed, the assertion made in the introduction of [13] that their groupoid is fairly far removed from the universal groupoid of S Λ might need rectification.…”

“…In both [25] and in [13] the relevant inverse semigroup is made to act on a topological space by means of partial homeomorphisms. The groupoid of germs for this action then turns out to be the appropriate groupoid.…”

Equivalent groupoids have Morita equivalent Steinberg algebras

“…Unlike [13] and [15], we do not require our graphs to be countable. More general versions are described in [9,12,18,26]. Define…”

Generalised morphisms of 𝑘-graphs: 𝑘-morphs

“…In particular, the higher-rank graphs and associated C * -algebras introduced in [18] have recently been widely studied [10,12,19,28]. Higher-rank graphs generalise directed graphs, so there are many points of similarity between the two theories, especially at the level of fundamental existence and uniqueness results.…”