Toeplitz algebras of semigroups

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