Row-finite Graph Research Articles

Maximal commutative subalgebras of Leavitt path algebras

Let [Formula: see text] be a field, and let [Formula: see text] be a row-finite (directed) graph. We present a construction of a wealth of maximal commutative subalgebras of the Leavitt path algebra [Formula: see text], which is a far-reaching generalization of the construction of the commutative core as a maximal commutative subalgebra of [Formula: see text].

Regular ideals of graph algebras

Let $C^*(E)$ be the graph C$^*$-algebra of a row-finite graph $E$. We give a complete description of the vertex sets of the gauge-invariant regular ideals of $C^*(E)$. It is shown that when $E$ satisfies Condition (L) the regular ideals $C^*(E)$ are a class of gauge-invariant ideals which preserve Condition (L) under quotients. That is, we show that if $E$ satisfies Condition (L) then a regular ideal $J \unlhd C^*(E)$ is necessarily gauge-invariant. Further, if $J \unlhd C^*(E)$ is a regular ideal, it is shown that $C^*(E)/J \simeq C^*(F)$ where $F$ satisfies Condition (L).

A note on the regular ideals of Leavitt path algebras

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition (L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.

Finiteness spaces, étale groupoids and their convolution algebras

Given a ring R, we extend Ehrhard’s linearization process by associating to any pre-finiteness space an R-module endowed with a Lefschetz topology. For a semigroup in the category of pre-finiteness spaces, one can endow this R-module with the convolution product to obtain an R-algebra. As examples of pre-finiteness spaces, we study topological spaces with bounded subsets (i.e., included in a compact) taken to be the finitary subsets. We prove that we obtain a finiteness space from any hemicompact space via this construction. As a corollary, any etale Hausdorff groupoid induces a semigroup in pre-finiteness spaces and its associated convolution algebra is complete in the hemicompact case. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.

The Non-selfadjoint Approach to the Hao–Ng Isomorphism

Abstract In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.

Classification of Leavitt Path Algebras with Two Vertices

We classify row-finite Leavitt path algebras associated to graphs with no more than two vertices. For the discussion we use the following invariants: decomposability, the $K_0$ group, $\det(N'_E)$ (included in the Franks invariants), the type, as well as the socle, the ideal generated by the vertices in cycles with no exits and the ideal generated by vertices in extreme cycles. The starting point is a simple linear algebraic result that determines when a Leavitt path algebra is IBN. An interesting result that we have found is that the ideal generated by extreme cycles is invariant under any isomorphism (for Leavitt path algebras whose associated graph is finite). We also give a more specific proof of the fact that the shift move produces an isomorphism when applied to any row-finite graph, independently of the field we are considering.

Derivations of Leavitt path algebras

In this paper, we describe the K-module HH1(LK(Γ)) of outer derivations of the Leavitt path algebra LK(Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of HH1(LK(Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C⁎-algebra.

Simplicity of the Lie Algebra of Skew Symmetric Elements of a Leavitt Path Algebra

For a field F of characteristic not 2 and a directed row-finite graph Γ let L(Γ) be the Leavitt Path Algebra with standard involution *. We study the lie algebra of K = K(L(Γ), *) of * −skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra [K, K] to be simple.

Centers of Path Algebras, Cohn and Leavitt Path Algebras

This paper is devoted to the study of the center of several types of path algebras associated to a graph E over a field K. First we consider the path algebra KE and prove that if the number of vertices is infinite then the center is zero; otherwise, it is K, except when the graph E is a cycle in which case the center is K[x], the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras. In the final section we describe the center of a prime graph C\(^*\)-algebra for a row-finite graph.

A groupoid generalisation of Leavitt path algebras

Let \(G\) be a locally compact, Hausdorff, etale groupoid whose unit space is totally disconnected. We show that the collection \(A(G)\) of locally-constant, compactly supported complex-valued functions on \(G\) is a dense \(*\)-subalgebra of \(C_c(G)\) and that it is universal for algebraic representations of the collection of compact open bisections of \(G\). We also show that if \(G\) is the groupoid associated to a row-finite graph or \(k\)-graph with no sources, then \(A(G)\) is isomorphic to the associated Leavitt path algebra or Kumjian–Pask algebra. We prove versions of the Cuntz–Krieger and graded uniqueness theorems for \(A(G)\).

TWO-SIDED IDEALS IN LEAVITT PATH ALGEBRAS

We explicitly describe two-sided ideals in Leavitt path algebras associated with a row-finite graph. Our main result is that any two-sided ideal I of a Leavitt path algebra associated with a row-finite graph is generated by elements of the form [Formula: see text], where g is a cycle based at vertex v. We use this result to show that a Leavitt path algebra is two-sided Noetherian if and only if the ascending chain condition holds for hereditary and saturated closures of the subsets of the vertices of the row-finite graph E.

The socle series of a Leavitt path algebra

We investigate the ascending Loewy socle series of Leavitt path algebras L K (E) for an arbitrary graph E and field K. We classify those graphs E for which L K (E) = S λ for some element S λ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of L K (E) is λ. Moreover, λ ≤ ω 1 (the first uncountable ordinal) if E is a row-finite graph.

Formula omitted]-algebras of separated graphs

The construction of the C ⁎ -algebra associated to a directed graph E is extended to incorporate a family C consisting of partitions of the sets of edges emanating from the vertices of E. These C ⁎ -algebras C ⁎ ( E , C ) are analyzed in terms of their ideal theory and K-theory, mainly in the case of partitions by finite sets. The groups K 0 ( C ⁎ ( E , C ) ) and K 1 ( C ⁎ ( E , C ) ) are completely described via a map built from an adjacency matrix associated to ( E , C ) . One application determines the K-theory of the C ⁎ -algebras U m , n n c , confirming a conjecture of McClanahan. A reduced C ⁎ -algebra C r e d ⁎ ( E , C ) is also introduced and studied. A key tool in its construction is the existence of canonical faithful conditional expectations from the C ⁎ -algebra of any row-finite graph to the C ⁎ -subalgebra generated by its vertices. Differences between C r e d ⁎ ( E , C ) and C ⁎ ( E , C ) , such as simplicity versus non-simplicity, are exhibited in various examples, related to some algebras studied by McClanahan.

AF-Embedding of crossed products of certain graph C*-algebras by quasi-free actions (II)

Let E be a row-finite directed graph, let G be a locally compact abelian group with dual group Ĝ = Γ, let ω be a labeling map from E* to Γ, and let (C*(E), G, α ω ) be the C*-dynamical system defined by ω. Some mappings concerning the AF-embedding construction of \(C*(E) \times _{\alpha ^\omega } G\) are studied in more detail. Several necessary conditions of AF-embedding and some properties of almost proper labeling map are obtained. Moreover it is proved that if E is constructed by attaching some 1-loops to a directed graph T consisting of some rooted directed trees and G is compact, then ω is almost proper, that is a sufficient condition for AF-embedding, if and only if Σ j=1 k \(\omega _{\gamma _j } \ne 1_\Gamma\) for any loop γ i , γ 2, ..., γ k attached to one path in T.

Realizations of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C ∗ -algebra, an Exel–Laca algebra, and an ultragraph C ∗ -algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C ∗ -algebras and Exel–Laca algebras, and that all simple AF-algebras are either graph C ∗ -algebras or Exel–Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C ∗ -algebra of a row-finite graph with no sinks.

The socle of a Leavitt path algebra

In this paper we characterize the minimal left ideals of a Leavitt path algebra as those which are isomorphic to principal left ideals generated by line points; that is, by vertices whose trees contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line point vertices. This characterization allows us to compute the socle of certain algebras that arise as the Leavitt path algebra of a row-finite graph. A complete description of the socle of a Leavitt path algebra is given: it is a locally matricial algebra.

Nonstable K-theory for Graph Algebras

We compute the monoid V(L K (E)) of isomorphism classes of finitely generated projective modules over certain graph algebras L K (E), and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of L K (E) and the lattice of order-ideals of V(L K (E)). When K is the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ is a dense subalgebra of the graph C *-algebra C *(E), and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.

Purely infinite simple Leavitt path algebras

We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L ( E ) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz–Krieger C ∗ -algebra C ∗ ( E ) given in [T. Bates, D. Pask, I. Raeburn, W. Szymański, The C ∗ -algebras of row-finite graphs, New York J. Math. 6 (2000) 307–324].

DYNAMICAL SYSTEMS IN GRAPH C*-ALGEBRAS

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism [Formula: see text]. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), [Formula: see text], hl(E) and hb(E) all coincide, where [Formula: see text] is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, [Formula: see text] has been known recently, and here we show that [Formula: see text], where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras [Formula: see text] of [Formula: see text] are examined to prove that [Formula: see text] for each vertex v.

The Leavitt path algebra of a graph

For any row-finite graph E and any field K we construct the Leavitt path algebra L ( E ) having coefficients in K. When K is the field of complex numbers, then L ( E ) is the algebraic analog of the Cuntz–Krieger algebra C ∗ ( E ) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings M n ( K ) and the Leavitt algebras L ( 1 , n ) appear as algebras of the form L ( E ) for various graphs E. In our main result, we give necessary and sufficient conditions on E which imply that L ( E ) is simple.