The noncommutative geometry of k-graph C*-algebras

Abstract

This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lowersemicontinuous gauge-invariant trace on C * (Λ) in terms of the existence of a faithful graph trace on Λ.Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k, ∞)summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the T k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.2. k-Graph C * -Algebras 2.1. Higher-rank graphs and their C * -algebras. In this subsection we outline the basic notation and definitions of k-graphs and their C * -algebras. We refer the reader to [RSY] for a more thorough account.Higher-rank graphs. Throughout this paper, we regard N k as a monoid under pointwise addition. We denote the usual generators of N k by e 1 , . . . , e k , and for n ∈ N k and 1 ≤ i ≤ k, we denote the i th coordinate of n by n i ∈ N; so n = n i e i . For m, n ∈ N k , we write m ≤ n if m i ≤ n i for all i. By m < n, we mean m ≤ n and m = n. We use m ∨ n and m ∧ n to denote, respectively, the coordinate-wise maximum and coordinate-wise minimum of m and n; so that m ∧ n ≤ m, n ≤ m ∨ n and these are respectively the greatest lower bound and least upper bound of m, n in N k . Definition 2.1 (Kumjian-Pask [KP]). A graph of rank k or k-graph is a pair (Λ, d) consisting of a countable category Λ and a degree functor d : Λ → N k which satisfy the following factorisation property: if λ ∈ Mor(Λ) satisfies d(λ) = m + n, then there are unique morphisms µ, ν ∈ Mor(Λ) such that d(µ) = m, d(ν) = n, and λ = µ • ν.The factorisation property ensures (see [KP]) that the identity morphisms of Λ are precisely the morphisms of degree 0; that is {id o : o ∈ Obj(Λ)} = d −1 (0). This means that we may identify each object with its identity morphism, and we do this henceforth. This done, we can regard Λ as consisting only of its morphisms, and we write λ ∈ Λ to mean λ ∈ Mor(Λ).Since we are thinking of Λ as a kind of graph, we write r and s for the codomain and domain maps of Λ respectively. We refer to elements of Λ as paths, and to the paths of degree 0 (which correspond to the objects of Λ as above) as vertices. Extending these conventions, we refer to the elements of Λ with minimal nonzero degree (that is d −1 ({e 1 , . . . , e k }) as edges.

Canonical Traces and Directly Finite Leavitt Path Algebras

“…This fact was noticed by Tomforde in [16] and later also utilized in [10] and [14]. Tomforde considers maps δ on vertices with values in (0, ∞) which satisfy the following two conditions and calls them graph traces on E.…”

“…Namely, in [14, Proposition 3.9], it has been shown that there is a bijective correspondence between faithful, C-valued graph traces on a countable, row-finite graph E and faithful, semifinite, lower semicontinuous, gauge invariant, C-valued traces on C * (E). In [14], a trace t on a C * -algebra A is defined as an additive map on the positive cone A + of A taking values in [0, ∞] such that t(ax) = at(x) for a nonnegative real number a and x ∈ A + and t(xx * ) = t(x * x) for all x ∈ A.…”

“…If {S e , p v | e ∈ E 1 , v ∈ E 0 } is a Cuntz-Krieger E-family for a graph C * -algebra C * (E) (see [7], [16] or [14] for example), the gauge action λ on the unit sphere S 1 is given by λ z (S p S * q ) = z |p|−|q| S p S * q for z ∈ S 1 and paths p and q. A C * -trace t on C * (E) is gauge invariant if t(λ z S p S * q ) = t(S p S * q ) for every complex number z of unit length.…”

“…Recently, Leavitt path algebras have joined this list as algebraic counterparts of graph C * -algebras and many properties of graph C * -algebras have been formulated for Leavitt path algebras and proven using solely algebraic methods. Our interest in traces is greatly inspired by their relevance in the study of noncommutative geometry of graph C * -algebras from [14].…”

“…[14,Proposition 3.9] shows that there is a bijective correspondence between the set of faithful, Cvalued graph traces on a countable, row-finite graph E and the set of faithful, semifinite, lower semicontinuous, gauge invariant, C-valued traces (in the operator theory sense) on the graph C * -algebra C * (E). We show that the two sets above are also in a bijective correspondence with the set of faithful, gauge invariant, C-linear, C-valued traces on the Leavitt path algebra L C (E) and that it is not necessary to require that E is row-finite (Corollary 3.8).…”