Cuntz Krieger Uniqueness Theorem Research Articles

QUOTIENTS OF ÉTALE GROUPOIDS AND THE ABELIANIZATIONS OF GROUPOID C*-ALGEBRAS

AbstractIn this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

The commutative core of a Leavitt path algebra

For any unital commutative ring R and for any graph E, we identify the commutative core of the Leavitt path algebra of E with coefficients in R, which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz–Krieger uniqueness theorem.

BRANCHING SYSTEMS FOR HIGHER-RANK GRAPH C*-ALGEBRAS

AbstractWe define branching systems for finitely aligned higher-rank graphs. From these, we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a generalized Cuntz–Krieger uniqueness theorem for periodic single-vertex 2-graphs. We use this result to give a sufficient condition under which representations of periodic single-vertex 2-graph C*-algebras arising from branching systems are faithful.

Uniqueness theorems for topological higher-rank graph 𝐶*-algebras

We consider the boundary-path groupoids of topological higher-rank graphs. We show that all such groupoids are topologically amenable. We deduce that the C ∗ C^* -algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz–Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph C ∗ C^* -algebra, and a condition under which it is also purely infinite.

Kumjian–Pask algebras of finitely aligned higher-rank graphs

We extend the definition of Kumjian–Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian–Pask algebras are universally defined and have a graded uniqueness theorem. We also prove the Cuntz–Krieger uniqueness theorem; to do this, we use a groupoid approach. As a consequence of the graded uniqueness theorem, we show that every Kumjian–Pask algebra is isomorphic to the Steinberg algebra associated to its boundary path groupoid. We then use Steinberg algebra results to prove the Cuntz–Krieger uniqueness theorem and also to characterize simplicity and basic simplicity.

The cycline subalgebra of a Kumjian-Pask algebra

Let $\Lambda$ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra $\mathcal{M}$ inside the Kumjian-Pask algebra ${\rm KP}_R(\Lambda)$. We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of ${\rm KP}_R(\Lambda)$ is injective if and only if it is injective on $\mathcal{M}$.

Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C*-algebras

We give a notion of branching systems on ultragraphs. From this, we build concrete representations of ultragraph [Formula: see text]-algebras on the bounded linear operators of Hilbert spaces. To each branching system of an ultragraph, we describe the associated Perron–Frobenius operator in terms of the induced representation. We show that every permutative representation of an ultragraph [Formula: see text]-algebra is unitary equivalent to a representation arising from a branching system. We give a sufficient condition on ultragraphs such that a large class of representations of the [Formula: see text]-algebras of these ultragraphs is permutative. To give a sufficient condition on branching systems, so that their induced representations are faithful, we generalize Szymański’s version of the Cuntz–Krieger uniqueness theorem to ultragraph [Formula: see text]-algebras.

Faithful Representations of Graph Algebras via Branching Systems

Abstract We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a suõcient condition under which the representation induced from a branching systemof a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We ûnish the paper by providing a proof of the converse of the Cuntz–Krieger uniqueness theorem for graph algebras by means of branching systems.

Leavitt R-algebras over countable graphs embed into L2,R

For a commutative ring R with unit we show that the Leavitt path algebra LR(E) of a graph E embeds into L2,R precisely when E is countable. Before proving this result we prove a generalised Cuntz–Krieger Uniqueness Theorem for Leavitt path algebras over R.

On freely generated semigraph C *-algebras

Abstract For special universal C*-algebras associated to k-semigraphs we present the universal representations of these algebras, prove a Cuntz–Krieger uniqueness theorem, and compute the K-theory. These C*-algebras seem to be the most universal Cuntz–Krieger like algebras naturally associated to k-semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient of such an algebra.

TWISTED TOPOLOGICAL GRAPH ALGEBRAS

We define the notion of a twisted topological graph algebra associated to a topological graph and a $1$-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological graph algebras. We prove a version of the Cuntz-Krieger uniqueness theorem, describe the gauge-invariant ideal structure. We find that a twisted topological graph algebra is simple if and only if the corresponding untwisted one is simple.

On twisted higher-rank graph 𝐶*-algebras

We define the categorical cohomology of a k k -graph Λ \Lambda and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k k -graph C ∗ C^* -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k k -graph C ∗ C^* -algebra is isomorphic to a twisted groupoid C ∗ C^* -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k k -graph C ∗ C^* -algebras are nuclear and belong to the bootstrap class.

Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics

Let R0 be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of R0, all of which have local units. We show that R0 is maximal commutative in the partial skew group ring R0⋊αG if and only if R0 has the ideal intersection property in R0⋊αG. From this we derive a criterion for simplicity of R0⋊αG in terms of maximal commutativity and G-simplicity of R0. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.

Crossed Products for Interactions and Graph Algebras

We consider Exel's interaction $(V,H)$ over a unital $C^*$-algebra $A$, such that $V(A)$ and $H(A)$ are hereditary subalgebras of $A$. For the associated crossed product, we obtain a uniqueness theorem, ideal lattice description, simplicity criterion and a version of Pimsner-Voiculescu exact sequence. These results cover the case of crossed products by endomorphisms with hereditary ranges and complementary kernels. As model examples of interactions not coming from endomorphisms we introduce and study in detail interactions arising from finite graphs. The interaction $(V,H)$ associated to a graph $E$ acts on the core $F_E$ of the graph algebra $C^*(E)$. By describing a partial homeomorphism of $\widehat{F}_E$ dual to $(V,H)$ we find Cuntz-Krieger uniqueness theorem, criteria for gauge-invariance of all ideals and simplicity of $C^*(E)$ as results concerning reversible noncommutative dynamics. We also provide a new approach to calculation of $K$-theory of $C^*(E)$ using only an induced partial automorphism of $K_0(F_E)$ and the six-term exact sequence.

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.

Kumjian–Pask algebras of locally convex higher-rank graphs

The Kumjian–Pask algebra of a higher-rank graph generalises the Leavitt path algebra of a directed graph. We extend the definition of Kumjian–Pask algebra to row-finite higher-rank graphs Λ with sources which satisfy a local-convexity condition. After proving versions of the graded-uniqueness theorem and the Cuntz–Krieger uniqueness theorem, we study the Kumjian–Pask algebra of rank-2 Bratteli diagrams by studying certain finite subgraphs which are locally convex. We show that the desourcification procedure of Farthing and Webster yields a row-finite higher-rank graph Λ˜ without sources such that the Kumjian–Pask algebras of Λ˜ and Λ are Morita equivalent. We then use the Morita equivalence to study the ideal structure of the Kumjian–Pask algebra of Λ by pulling the appropriate results across the equivalence.

A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k-graph C⁎-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph C⁎-algebra, it is sufficient that the representation be injective on a distinguished abelian C⁎-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C⁎-algebra, each of which is the unique extension of a state on the distinguished abelian C⁎-subalgebra.

Simple Cuntz–Pimsner rings

Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz–Pimsner ring contains a non-zero graded ideal, when a relative Cuntz–Pimsner ring is simple, and when every ideal in a relative Cuntz–Pimsner ring is graded, are given. A “Cuntz–Krieger uniqueness theorem” for relative Cuntz–Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz–Pimsner rings are introduced.As applications of these results, a uniqueness result for the Toeplitz algebra of a directed graph and characterizations of when crossed products of a ring by a single automorphism and fractional skew monoid rings of a single corner isomorphism are simple, are obtained.

Abelian core of graph algebras

We introduce a certain natural abelian C*-subalgebra of a graph C*-algebra, which has functorial properties that can be used for characterizing injectivity of representations of the ambient C*-algebra. In particular, a short proof of the Cuntz-Krieger Uniqueness Theorem, for graphs that may have loops without entries, is given. Graph C*-algebras are interesting objects with a structure that can often be understood in terms of the combinatorics and geometry of the underlying graph. A large class of examples include AF-algebras, Cuntz-Krieger algebras, and C*algebras that are built up from matrices over C(T). Here we consider C*-algebras defined from countable directed graphs. In particular, such a graph is determined by a four-tuple E = (E, E, r, s) consisting of its vertex set, edge set, and range and source maps. Given such a graph E we define a Cuntz-Krieger E-system on a Hilbert space H to be a collection {Sμ |μ ∈ E ∪ E} of operators on H, where Sμ is an orthogonal projection whenever μ ∈ E and a partial isometry whenever μ ∈ E, and the usual Cuntz-Krieger conditions are satisfied: (CK 1) S∗ eSe = Ss(e) for every e ∈ E (CK 2) Sv = ∑ r(e)=v SeS ∗ e whenever {e ∈ E | r(e) = v} is nonempty and finite. In light of the restriction on the vertex v in condition (CK 2) we need also to require that SeS ∗ e ≤ Sr(e) for every e ∈ E, and that the projections SeS e are mutually orthogonal. Let us call such a system nondegenerate if there is no vertex v ∈ E with Pv = 0. Note that our convention for the initial and range spaces of the partial isometries agrees with [R]. In the usual way, one can prove the existence of a universal C*-algebra generated by a Cuntz-Krieger E system; we denote this algebra by C∗〈E〉. By a uniqueness theorem one means a set of conditions on the graph E or on the map π guaranteeing that a representation π : C∗〈E〉 → A is injective. The first of the Cuntz-Krieger uniqueness theorems was Coburn’s Theorem (see, e.g., [Mu]), which asserts that any C*-algebra generated by a nonunitary partial isometry is isomorphic to the Toeplitz Algebra T . Taking the graph E with E = {v, w} and E = {e, f}, where s(f) = v and s(e) = r(e) = r(f) = w, one can verify that in any E-system the operator Se + Sf is a nonunitary partial isometry, and thus C ∗〈E〉 = T . Ten years later, in [C], Cuntz showed that when E consists of a single vertex and n edges (loops), any algebra generated by an E-system is isomorphic to the Cuntz algebra On ([CK]). Subsequently, Cuntz and Krieger proved that the graph with adjacency matrix A 1991 Mathematics Subject Classification. Primary 46L10; Secondary 46L30.

Leavitt path algebras with coefficients in a commutative ring

Given a directed graph E we describe a method for constructing a Leavitt path algebra L R ( E ) whose coefficients are in a commutative unital ring R . We prove versions of the Graded Uniqueness Theorem and Cuntz–Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of L R ( E ) , and we prove that if K is a field, then L K ( E ) ≅ K ⊗ Z L Z ( E ) .