Howe–Moore type theorems for quantum groups and rigid -tensor categories

We formulate and study Howe–Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast }$-tensor categories. We say that a rigid $C^{\ast }$-tensor category ${\mathcal{C}}$ has the Howe–Moore property if every completely positive multiplier on ${\mathcal{C}}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe–Moore property. As an immediate consequence, we deduce the Howe–Moore property for Temperley–Lieb–Jones standard invariants with principal graph $A_{\infty }$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan–Wenzl categories. Additionally, in the specific case of the quantum groups $\text{SU}_{q}(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\text{SU}_{q}(N)$, which coincide with the completely positive multipliers on the representation category of $\text{SU}_{q}(N)$.