We introduce and analyze the full $\mathcal{ NT}_{ \mathcal{ L}}(\mathcal{ K})$ and the reduced $\mathcal{ NT}_{ \mathcal{ L}}^r(\mathcal{ K})$ Nica-Toeplitz algebra associated to an ideal $\mathcal{ K}$ in a right tensor $C^*$-precategory $\mathcal{ L}$ over a right LCM semigroup $P$. Our main results are uniqueness theorems in the spirit of classical Coburn's theorem, generalizing uniqueness results for Toeplitz-type $C^*$-algebras associated to single $C^*$-correspondences, quasi-lattice ordered semigroups, and crossed products twisted by product systems of $C^*$-correspondences obtained by Fowler, Laca and Raeburn. We formulate geometric conditions on a representation $\Phi$ of $\mathcal{ K}$ so that the $C^*$-algebra it generates, $C^*(\Phi(\mathcal{ K}))$, naturally lies between $\mathcal{ NT}_{ \mathcal{ L}}^r(\mathcal{ K})$ and $\mathcal{ NT}_{ \mathcal{ L}}(\mathcal{ K})$. Under suitable amenability hypotheses, $C^*(\Phi(\mathcal{ K}))$ and $\mathcal{ NT}_{ \mathcal{ L}}(\mathcal{ K})$ are isomorphic. The geometric conditions are necessary for our uniqueness result when the right tensoring preserves $\mathcal{ K}$ and in general they capture uniqueness of the $C^*$-algebra generated by a natural extension of $\Phi$ to $\mathcal{ L}$. In particular, the latter algebra could be viewed as a Doplicher-Roberts version of $\mathcal{ NT}_{ \mathcal{ L}}(\mathcal{ K})$.
Categories: math.OA