Cohn path algebras have Invariant Basis Number
For any finite directed graph $E$ and any field $K$ we show that the Cohn path algebra $C_K(E)$ has the Invariant Basis Number property.
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arXiv subjects
Gene Abrams
Publications and source records attributed to Gene Abrams.
The Leavitt path algebras of generalized Cayley graphs
Let $n$ be a positive integer. For each $0\leq j \leq n-1$ we let $C_n^{j}$ denote Cayley graph for the cyclic group ${\mathbb Z}_n $ with respect to the subset $\{1, j\}$. For any such pair $(n,j)$ we compute the size of the Grothendieck group of the Leavitt path algebra $L_K(C_n^j)$; the analysis is related to a collection of integer sequences described by Haselgrove in the 1940's. When $j=0,1,$ or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras $L_K(C_n^j)$ as the Leavitt path algebras of graphs having at most three vertices. The analysis in the $j=2$ case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.
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A class of C*-algebras that are prime but not primitive
We establish necessary and sufficient conditions on a (not necessarily countable) graph E for the graph C*-algebra C*(E) to be primitive. Along with a known characterization of the graphs E for which C*(E) is prime, our main result provides us with a systematic method for easily producing large classes of (necessarily nonseparable) C*-algebras that are prime but not primitive. We also compare and contrast our results with similar results for Leavitt path algebras.
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Simple Lie algebras arising from Leavitt path algebras
For a field K and directed graph E, we analyze those elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E), L_K(E)]. This analysis allows us to give easily computable necessary and sufficient conditions to determine which Lie algebras of the form [L_K(E), L_K(E)] are simple, when E is row-finite (i.e., has finite out-degree) and L_K(E) is simple.
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On prime non-primitive von Neumann regular algebras
Let $E$ be any directed graph, and $K$ any field. We classify those graphs $E$ for which the Leavitt path algebra $L_K(E)$ is primitive. As a consequence, we obtain classes of examples of von Neumann regular prime rings which are not primitive.
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Row-finite equivalents exist only for row-countable graphs
If $E$ is a not-necessarily row-finite graph, such that each vertex of $E$ emits at most countably many edges, then a {\it desingularization} $F$ of $E$ can be constructed (see e.g. (1) G. Abrams, G. Aranda Pino, Leavitt path algebras of arbitrary graphs, Houston J. Math 34(2) (2008), 423-442, or (2) I. Raeburn, "Graph algebras". CBMS Regional Conference Series in Mathematics 103, Conference Board of the Mathematical Sciences, Washington, DC, 2005, ISBN 0-8218-3660-9). The desingularization process has been effectively used to establish various characteristics of the Leavitt path algebras of not-necessarily row-finite graphs. Such a desingularization $F$ of $E$ has the properties that: (1) $F$ is row-finite, and (2) the Leavitt path algebras $L(E)$ and $L(F)$ are Morita equivalent. We show here that for an arbitrary graph $E$, a graph $F$ having properties (1) and (2) exists (we call such a graph $F$ a \emph{row-finite equivalent of} $E$) if and only if $E$ is row-countable; that is, $E$ contains no vertex $v$ for which $v$ emits uncountably many edges.
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The Dixmier-Moeglin equivalence for Leavitt path algebras
Let $K$ be a field, let $E$ be a finite directed graph, and let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$. We show that for a prime ideal $P$ in $L_K(E)$, the following are equivalent: \begin{enumerate} \item $P$ is primitive; \item $P$ is rational; \item $P$ is locally closed in ${\rm Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum ${\rm Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to ${\rm Spec}(K)$ or to ${\rm Spec}(K[x,x^{-1}])$. In the case that $K$ is infinite, we show that $L_K(E)$ has a rational $K^{\times}$-action, and that the indicated decomposition of ${\rm Spec}(L_K(E))$ is induced by this action.
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On the simplicity of Lie algebras associated to Leavitt algebras
For any field $\K$ and integer $n\geq 2$ we consider the Leavitt algebra $L_\K(n)$; for any integer $d\geq 1$ we form the matrix ring $S = M_d(L_\K(n))$. $S$ is an associative algebra, but we view $S$ as a Lie algebra using the bracket $[a,b]=ab-ba$ for $a,b \in S$. We denote this Lie algebra as $S^-$, and consider its Lie subalgebra $[S^-,S^-]$. In our main result, we show that $[S^-,S^-]$ is a simple Lie algebra if and only if char$(\K)$ divides $n-1$ and char$(\K)$ does not divide $d$. In particular, when $d=1$ we get that $[L_\K(n)^-,L_\K(n)^-]$ is a simple Lie algebra if and only if char$(\K)$ divides $n-1$.
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Isomorphism and Morita equivalence of graph algebras
For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$. We prove that in certain situations isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ in specific cases. The relationship between Leavitt path algebras and graph C*-algebras is also explored in the context of Morita equivalence.
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The matrix type of purely infinite simple Leavitt path algebras
Let $R$ denote the purely infinite simple unital Leavitt path algebra $L(E)$. We completely determine the pairs of positive integers $(c,d)$ for which there is an isomorphism of matrix rings $M_c(R)\cong M_d(R)$, in terms of the order of $[1_R]$ in the Grothendieck group $K_0(R)$.
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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, $λ\leq ω$ (the first infinite ordinal) if $E$ is a row-finite graph.
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