## Semigroup crossed products and the Toeplitz algebras of nonabelian groups.(English)Zbl 0887.46040

$$P$$ being a subsemigroup of a group $$G$$, A. Nica defined in [J. Oper. Theory 27, No. 1, 17-52 (1992; Zbl 0809.46058)] quasilattice ordered groups $$(G,P)$$ and covariant representations of the dynamical system $$(B_P,P,\alpha)$$, where $$\alpha$$ is a natural action of $$P$$ on a $$C^*$$-subalgebra $$B_P$$ of $$\ell^\infty(P)$$. This dynamical system can be identified with the universal Toeplitz algebra of $$(G,P)$$ and it is shown that its Toeplitz representation is faithful if and only if $$(G,P)$$ satisfies an amenability condition, a Coburn’s type uniqueness theorem. Then free products of Abelian quasilattice orders are shown to be amenable. Hence results of Cuntz and Dinh on the uniqueness of Toeplitz-Cuntz algebras are extended to this context.

### MSC:

 46L55 Noncommutative dynamical systems

Zbl 0809.46058
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