Semigroup Crossed Products and the Toeplitz Algebras of Nonabelian Groups

We consider the quasi-lattice ordered groups (G, P) recently introduced by Nica. We realise their universal Toeplitz algebra as a crossed productBP⋊Pby a semigroup of endomorphisms, and show that the Toeplitz representation is faithful precisely when an amenability condition is satisfied. We then show directly that many interesting free products are amenable. The final result contains theorems of Cuntz and Dinh on the uniqueness of Toeplitz–Cuntz algebras as special cases.