## Amalgamation of inverse semigroups and operator algebras

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##### Abstract

We focus on three constructions: amalgamated free products of inverse semi-groups, C*-algebras of inverse semigroups, and amalgamated free products of C*-algebras. The starting point is an amalgam [S1, S2, U] of inverse semigroups that is full, i.e., the embeddings of U into S1 and S2 are bijective on the semilattice of idempotents. Although the order structure of the amalgamated free product is well-understood, the structure of the maximal subgroups was somewhat mysterious prior to this work. We use Bass-Serre theory to characterize these maximal subgroups and determine which graphs of groups arise in this setting. We obtain necessary and sufficient conditions for the amalgamated free product to have trivial subgroups. One surprising consequence is that an amalgamated free product of finite inverse semigroups may be finite. We analyze the structure of the C*-algebra of an inverse monoid S using techniques developed by Sieben. Let E be the semilattice of idempotents of S, and extend the Munn action of S on E to a partial action of S on C*(E). We prove that C*(S) is isomorphic to the partial crossed product of C* (E) and S using this action. To generalize our construction to inverse semigroups, we determine the effect on C*(S) of attaching an identity to S. Our construction simplifies the construction given by Paterson. Finally we consider C* (S) when the inverse semigroup S is the amalgamated free product of a full amalgam [S1, S2, U]. We prove that the C*-functor commutes with the formation of amalgamated free products under this hypothesis. We prove an analogous result for the complex algebra of S. Using the characterization of maximal subgroups given above, we identify some amalgamated free products of C*-algebras by recognizing them as C*-algebras of inverse semigroups. Thus, we can identify certain amalgams whose K-theory was found by McClanahan.