Amalgams of inverse semigroups and $C^*$-algebras Allan DonsigSteven HaatajaJohn Meakin 46L0920M20$C^*$-algebrainverse semigroupamalgamated free product 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 $[S,T,U]$, $C^{*}(S*_UT) \cong C^{*}(S)*_{C^{*}(U)}C^{*}(T)$. 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. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4255 10.1512/iumj.2011.60.4255 en Indiana Univ. Math. J. 60 (2011) 1059 - 1076 state-of-the-art mathematics http://iumj.org/access/