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Uniqueness theorems and ideal structure for Leavitt path algebras
, 2008
"... We prove Leavitt path algebra versions of the two uniqueness theorems of graph C ∗algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the ..."
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Cited by 58 (7 self)
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We prove Leavitt path algebra versions of the two uniqueness theorems of graph C ∗algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra LC(E) embeds as a dense ∗subalgebra of the graph C ∗algebra C ∗ (E). This embedding has consequences for graph C ∗algebras, and we discuss how we obtain new information concerning the construction of C ∗ (E).
The classification question for Leavitt path algebras
 J. Algebra
, 2008
"... Abstract. We prove an algebraic version of the GaugeInvariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between Zgraded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified ..."
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Cited by 24 (10 self)
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Abstract. We prove an algebraic version of the GaugeInvariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between Zgraded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the K0 groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in K0, classify the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using numbertheoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras.
ISOMORPHISM AND MORITA EQUIVALENCE OF GRAPH ALGEBRAS
, 2008
"... For any countable graph E, we investigate the relationship between the Leavitt path algebra LC(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 LC(E) and LC(F). We prove that in cer ..."
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Cited by 21 (7 self)
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For any countable graph E, we investigate the relationship between the Leavitt path algebra LC(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 LC(E) and LC(F). We prove that in certain situations isomorphisms between LC(E) and LC(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 LC(E) and LC(F) in specific cases. The relationship between Leavitt path algebras and graph C ∗algebras is also explored in the context of Morita equivalence.
Regularity conditions for arbitrary Leavitt path algebras
 ALGEBRAS AND REPRESENTATION THEORY
, 2008
"... We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally Kmatricial; that is, LK(E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality condi ..."
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Cited by 17 (11 self)
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We show that if E is an arbitrary acyclic graph then the Leavitt path algebra LK(E) is locally Kmatricial; that is, LK(E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graphtheoretic conditions (e.g. rowfiniteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) LK(E) is von Neumann regular. (2) LK(E) is πregular. (3) E is acyclic. (4) LK(E) is locally Kmatricial. (5) LK(E) is strongly πregular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.
LEAVITT PATH ALGEBRAS WITH COEFFICIENTS IN A COMMUTATIVE RING
, 2009
"... Given a directed graph E we describe a method for constructing a Leavitt path algebra LR(E) whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and CuntzKrieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both genera ..."
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Cited by 12 (3 self)
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Given a directed graph E we describe a method for constructing a Leavitt path algebra LR(E) whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and CuntzKrieger 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 LR(E), and we prove that if K is a field, then LK(E) ∼ = K ⊗Z LZ(E) if the character of K is 0, and LK(E) ∼ = K ⊗Zp LZp(E) if the character of K is a prime p.
PRIME SPECTRUM AND PRIMITIVE LEAVITT PATH ALGEBRAS
, 2008
"... Abstract. In this paper a bijection between the set of prime ideals of a Leavitt path algebra LK(E) and a certain set which involves maximal tails in E and the prime spectrum of K[x, x −1] is established. Necessary and sufficient conditions on the graph E so that the Leavitt path algebra LK(E) is pr ..."
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Cited by 12 (3 self)
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Abstract. In this paper a bijection between the set of prime ideals of a Leavitt path algebra LK(E) and a certain set which involves maximal tails in E and the prime spectrum of K[x, x −1] is established. Necessary and sufficient conditions on the graph E so that the Leavitt path algebra LK(E) is primitive are also found. introduction Leavitt path algebras of rowfinite graphs have been recently introduced in [1] and [7]. They have become a subject of significant interest, both for algebraists and for analysts working in C*algebras. The CuntzKrieger algebras C ∗ (E) (the C*algebra counterpart of these Leavitt path algebras) are described in [21]. The algebraic and analytic theories, while sharing some striking similarities, they present some remarkable differences, as was shown for instance in the “Workshop on Graph Algebras ” held at the University of Málaga (see [11]), and more deeply in the subsequent enlightening work of Tomforde [23]. For a field K, the algebras LK(E) are natural generalizations of the algebras investigated by Leavitt in [19], and are a specific type of path Kalgebras associated to a graph E (modulo certain relations). The family of algebras which can be realized as the Leavitt path algebras
THE SOCLE SERIES OF A LEAVITT PATH ALGEBRA
, 2009
"... We investigate the ascending Loewy socle series of Leavitt path algebras LK(E) for an arbitrary graph E and field K. We classify those graphs E for which LK(E) = Sλ for some element Sλ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of ..."
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Cited by 6 (3 self)
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We investigate the ascending Loewy socle series of Leavitt path algebras LK(E) for an arbitrary graph E and field K. We classify those graphs E for which LK(E) = Sλ for some element Sλ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of LK(E) is λ. Moreover, λ ≤ ω (the first infinite ordinal) if E is a rowfinite graph.
WEAKLY REGULAR AND SELFINJECTIVE LEAVITT PATH ALGEBRAS OVER ARBITRARY GRAPHS
"... Abstract. We characterize the Leavitt path algebras over arbitrary graphs which are weakly regular rings as well as those which are selfinjective. In order to reach our goals we extend and prove several results on projective, injective and flat modules over Leavitt path algebras and, more generally ..."
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Cited by 6 (4 self)
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Abstract. We characterize the Leavitt path algebras over arbitrary graphs which are weakly regular rings as well as those which are selfinjective. In order to reach our goals we extend and prove several results on projective, injective and flat modules over Leavitt path algebras and, more generally, over (not necessarily unital) rings with local units. 1. Introduction and
On prime nonprimitive von Neumann regular algebras
 Trans. Amer. Math. Soc
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