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 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 K 0 groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in K 0 , 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 number-theoretic methods, can be reobtained via appropriate isomorphisms between 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.

Proof. Clearly, the proposed comultiplication and counit are homomorphisms of A ⊗ k Bbimodules. To check the coassociative property, consider the diagram (C ⊗ A

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