A universal property of semigroup $C^*$-algebras for free products of semigroups of rationals

Abstract

The paper deals with the left reduced semigroup $C^*$-algebras $C^*_\lambda(Q)$ for non-abelian cancellative semigroups $Q$ associated with finite tuples $(M_1,M_2, \ldots , M_n)$ of sequences $M_i$ of arbitrary natural numbers. Such a semigroup $Q$ is defined to be the free  product  of semigroups consisting of positive numbers  in the ordered groups of rationals $Q_{M_i}$ generated by the reciprocals for products of the terms in $M_i$. The $C^*$-algebra $C^*_\lambda(Q)$ is generated by the left regular representation of $Q$. It is shown  that each semigroup $Q$ is not left amenable but its full and left reduced semigroup $C^*$-algebras are isomorphic and nuclear. We establish that every $C^*$-algebra $C^*_\lambda(Q)$ can be characterized as a universal $C^*$-algebra defined by a countable set of isometries satisfying a countable family of polynomial relations.  To prove this result, we make use of the universal property of   $C^*$-algebra  $C^*_\lambda(Q)$ considered as the inductive limit for the inductive sequence of   Toeplitz — Cuntz algebras associated with the tuple $(M_1,M_2, \ldots , M_n)$.