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.


46L55 Noncommutative dynamical systems


Zbl 0809.46058
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