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48
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).
Purely infinite simple Leavitt path algebras
, 2005
"... We give necessary and sufficient conditions on a rowfinite 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 CuntzKrieger C∗algebra C ∗ (E) given in [7]. ..."
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Cited by 37 (17 self)
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We give necessary and sufficient conditions on a rowfinite 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 CuntzKrieger C∗algebra C ∗ (E) given in [7].
FLOW INVARIANTS IN THE CLASSIFICATION OF LEAVITT PATH ALGEBRAS
, 2009
"... We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two ..."
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Cited by 17 (4 self)
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We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two main results: the first gives sufficient conditions for which the Leavitt path algebras in a certain class are Morita equivalent, while the second gives sufficient conditions which yield isomorphisms. We discuss a possible approach to establishing whether or not these conditions are also in fact necessary. In the final section we present many additional operations on graphs which preserve Morita equivalence (resp., isomorphism) of the corresponding Leavitt path algebras.
Module theory over Leavitt path algebras and
 Ktheory, Preprint. PERE ARA AND MIQUEL BRUSTENGA
"... Abstract. Let k be a field and let E be a finite quiver. We study the structure of the finitely presented modules of finite length over the Leavitt path algebra Lk(E) and show its close relationship with the finitedimensional representations of the inverse quiver E of E, as well as with the class o ..."
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Cited by 12 (3 self)
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Abstract. Let k be a field and let E be a finite quiver. We study the structure of the finitely presented modules of finite length over the Leavitt path algebra Lk(E) and show its close relationship with the finitedimensional representations of the inverse quiver E of E, as well as with the class of finitely generated Pk(E)modules M such that Tor Pk(E) q (kE0, M) = 0 for all q, where Pk(E) is the usual path algebra of E. By using these results we compute the higher Ktheory of the von Neumann regular algebra Qk(E) = Lk(E)Σ−1, where Σ is the set of all square matrices over Pk(E) For a field k and an integer n ≥ 2, the Leavitt algebra L(1, n) of type (1, n) is the algebra with generators xi, yj, 1 ≤ i, j ≤ n with defining relations given by (x1,...,xn)(y1,...,yn) t = 1,
THE REALIZATION PROBLEM FOR VON NEUMANN REGULAR RINGS
, 2008
"... We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right Rmodules over a von Neumann regular ring R. ..."
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Cited by 11 (7 self)
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We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right Rmodules over a von Neumann regular ring R.
The ranges of Ktheoretical invariants for nonsimple graph algebras
 In preparation
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