Flow Invariants in the Classification of Leavitt Path Algebras

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. Introduction Throughout this article E will denote a row-finite directed graph, and K will denote an arbitrary field. The Leavitt path algebra of E with coefficients in K, denoted LK(E), has received significant attention over the past few years, both from algebraists as well as from analysts working in operator theory. (The precise definition of LK(E) is given below.) When K is the field C of complex numbers, the algebra LK(E) has exhibited surprising similarity to C(E), the graph C-algebra of E. In this context, it is natural to ask whether an analog of the Kirchberg-Phillips Classification Theorem [13, 18] for C-algebras holds for various classes of Leavitt path algebras as well. Specifically, the following question was posed in [4]: The Classification Question for purely infinite simple unital Leavitt path algebras: Let K be a field, and suppose E and F are graphs for which LK(E) and LK(F ) are purely infinite simple unital. If K0(LK(E)) ∼= K0(LK(F )) via an isomorphism φ having φ([1LK(E)]) = [1LK(F )], must LK(E) and LK(F ) be isomorphic? The Classification Question is answered in the affirmative in [4] for a few specific classes of graphs. We obtain in the current article an affirmative answer for a significantly wider class of graphs. Our approach is as follows. In Section 1 we consider Morita equivalence of 2000 Mathematics Subject Classification. Primary 16D70, Secondary 46L05.