Toeplitz algebras of semigroups

To each monoid $P$ that embeds in a group we associate a universal Toeplitz C*-algebra $T_u(P)$ defined via generators and relations; $T_u(P)$ is a quotient of Li's semigroup C*-algebra $C^*(P)$ and they are isomorphic if and only if $P$ satisfies independence. We give a partial crossed product realization of $T_u(P)$ and show that several key results known for $C^*(P)$ when $P$ satisfies independence are also valid for $T_u(P)$ when independence fails. At the level of the reduced semigroup C*-algebra $T_r(P)$, we show that nontrivial ideals have nontrivial intersection with the reduced crossed product of the diagonal subalgebra by the action of the group of units of $P$, generalizing a result of Li for monoids with trivial unit group. We also characterize when the action of the group of units is topologically free and we show that in this case a representation of $T_r(P)$ is faithful iff it is jointly proper. This yields a uniqueness theorem for C*-algebras generated by semigroups of isometries that unifies several classical results. We provide a presentation for the covariance algebra of the product system over $P$ with one-dimensional fibers in terms of a notion of foundation sets of constructible ideals that generalizes that of Sims and Yeend for quasi-lattice orders. The covariance algebra is a full, or universal, analogue of the boundary quotient. We give purely algebraic sufficient conditions on $P$ for the boundary quotient to be purely infinite simple, which reduce to Starling's conditions in the case of right LCM monoids. We discuss applications of our results to examples that include a numerical semigroup and the $ax+b$-monoid of an integral domain. This is particularly interesting in the case of nonmaximal orders in number fields, for which we show independence always fails. In addition, we simplify and generalize results for right LCM monoids.