Amalgams of inverse semigroups and $C^*$-algebras

Abstract

An amalgam of inverse semigroups [S, T, U ] is full if U contains all of the idempotents of S and T . We show that for a full amalgam [S, T, U ], C * (S * U T ) ∼ = C * (S) * C * (U) C * (T ). Using this result, we describe certain amalgamated free products of C * -algebras, including finite-dimensional C * -algebras, the Toeplitz algebra, and the Toeplitz C * -algebras of graphs. 2010 Mathematics Subject Classification. 46L09, 20M20.

A tribute to John Meakin on the occasion of his 75th birthday

“…There has been an immense amount of work on C * -algebras of inverse semigroups and related groupoids over the last few decades. John, Donsig and our late student Steven Haataja studied the relationship between amalgams of inverse semigroups and C * -algebras in [12]. The papers [47,53] are concerned with inverse semigroups and Leavitt path algebras and graph inverse semigroups respectively.…”