# Isomorphisms of algebras associated with directed graphs

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## Abstract.

Given countable directed graphs *G* and *G*′, we show that the associated tensor algebras Open image in new window(*G*) and Open image in new window(*G*′) are isomorphic as Banach algebras if and only if the graphs *G* are *G*′ are isomorphic. For tensor algebras associated with graphs having no sinks or no sources, the graph forms an invariant for algebraic isomorphisms. We also show that given countable directed graphs *G*, *G*′, the free semigroupoid algebras Open image in new window and Open image in new window are isomorphic as dual algebras if and only if the graphs *G* are *G*′ are isomorphic. In particular, spatially isomorphic free semigroupoid algebras are unitarily isomorphic. For free semigroupoid algebras associated with locally finite directed graphs with no sinks, the graph forms an invariant for algebraic isomorphisms as well.

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