Nica-Toeplitz algebras associated with right-tensor C*-precategories over right LCM semigroups

We introduce and analyze the full NTL(K) and the reduced NTLr(K) Nica-Toeplitz algebra associated to an ideal K in a right-tensor C*-precategory L over a right LCM semigroup P. These C*-algebras unify cross-sectional C*-algebras associated to Fell bundles over discrete groups and Nica-Toeplitz C*-algebras associated to product systems. They also allow a study of Doplicher-Roberts versions of the latter. A new phenomenon is that when P is not right cancellative then the canonical conditional expectation takes values outside the ambient algebra. Our main result is a uniqueness theorem that gives sufficient conditions for a representation of K to generate a C*-algebra naturally lying between NTL(K) and NTLr(K). We also characterize the situation when NTL(K) congruent to NTLr(K). Unlike previous results for quasi-lattice monoids, P is allowed to contain nontrivial invertible elements, and we accommodate this by identifying an assumption of aperiodicity of an action of the group of invertible elements in P. One prominent condition for uniqueness is a geometric condition of Coburn's type, exploited in the work of Fowler, Laca and Raeburn. Here we shed new light on the role of this condition by relating it to a C*-algebra associated to L itself.