Flow invariants in the classification of Leavitt path algebras

@article{Abrams2008FlowII,
  title={Flow invariants in the classification of Leavitt path algebras},
  author={Gene Abrams and A. Louly and Enrique Pardo and Connor Smith},
  journal={arXiv: Rings and Algebras},
  year={2008}
}
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… Expand
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References

SHOWING 1-10 OF 40 REFERENCES
Classification Question for Leavitt Path Algebras
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between Z-graded algebras. As our mainExpand
The Leavitt path algebra of a graph
Abstract 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 algebraicExpand
Purely infinite simple Leavitt path algebras
Abstract 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 theExpand
Flow equivalence of graph algebras
This paper explores the effect of various graphical constructions upon the associated graph C*-algebras. The graphical constructions in question arise naturally in the study of flow equivalence forExpand
Nonstable K-theory for Graph Algebras
We compute the monoid V(LK(E)) of isomorphism classes of finitely generated projective modules over certain graph algebras LK(E), and we show that this monoid satisfies the refinement property andExpand
Exchange Leavitt path algebras and stable rank
Abstract We characterize those Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 orExpand
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 Mc(R) ≌Expand
Finite-dimensional Leavitt path algebras
Abstract We classify the directed graphs E for which the Leavitt path algebra L ( E ) is finite dimensional. In our main results we provide two distinct classes of connected graphs from which, moduloExpand
Automorphisms of Bowen–Franks groups of shifts of finite type
  • Danrun Huang
  • Mathematics
  • Ergodic Theory and Dynamical Systems
  • 2001
There are four Bowen–Franks groups associated to each shift of finite type. For an irreducible shift of finite type, we show that a 4-tuple of automorphisms corresponding to the four Bowen–FranksExpand
A class ofC*-algebras and topological Markov chains
In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from aExpand
...
1
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3
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