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

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