Let P be a unital subsemigroup of a group G. We propose an approach to C * -algebras associated to product systems over P . We call the C * -algebra of a given product system E its covariance algebra and denote it by A× E P , where A is the coefficient C * -algebra. We prove that our construction does not depend on the embedding P → G and that a representation of A × E P is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz-Nica-Pimsner algebras, Fowler's Cuntz-Pimsner algebra, semigroup C * -algebras of Xin Li and Exel's crossed products by interaction groups.

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We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute Cuntz-Pimsner algebras of proper correspondences as colimits in the correspondence bicategory.

To each submonoid P P of a group we associate a universal Toeplitz C ∗ \mathrm {C}^* -algebra T u ( P ) \mathcal {T}_u(P) defined via generators and relations; T u ( P ) \mathcal {T}_u(P) is a quotient of Li’s semigroup C ∗ \mathrm {C}^* -algebra C s ∗ ( P ) \mathrm {C}^*_s(P) and they are isomorphic iff P P satisfies independence. We give a partial crossed product realization of T u ( P ) \mathcal {T}_u(P) and show that several results known for C s ∗ ( P ) \mathrm {C}^*_s(P) when P P satisfies independence are also valid for T u ( P ) \mathcal {T}_u(P) when independence fails. At the level of the reduced semigroup C ∗ \mathrm {C}^* -algebra T λ ( P ) \mathcal {T}_\lambda (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 P , generalizing a result of Li for monoids with trivial unit group. We characterize when the action of the group of units is topologically free, in which case a representation of T λ ( P ) \mathcal {T}_\lambda (P) is faithful iff it is jointly proper. This yields a uniqueness theorem that generalizes and unifies several classical results. We provide a concrete presentation for the covariance algebra of the product system over P P with one-dimensional fibers in terms of a new notion of foundation sets of constructible ideals. We show that the covariance algebra is a universal analogue of the boundary quotient and give conditions on  P P for the boundary quotient to be purely infinite simple. We discuss applications to a numerical semigroup and to the a x + b 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.

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