Continuous Tambara-Yamagami tensor categories

We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of $\mathrm{C}^*$-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups to locally compact abelian groups, and provide a classification of continuous Tambara-Yamagami tensor categories for a locally compact group $G$. A continuous Tambara-Yamagami tensor category associated to a locally compact group $G$ is a continuous tensor category that has a single non-invertible simple object $\tau$ such that $\tau\otimes \tau$ decomposes as a direct integral indexed over $G$, meaning $\tau\otimes\tau \cong L^2(G)$. We show that continuous Tambara-Yamagami tensor categories for $G$ are classified by a continuous symmetric nondegenerate bicharacter $\chi: G\times G\to U(1)$ and a sign $\xi\in\{\pm 1\}$. We also prove that, if a $\mathrm{W}^*$-tensor category $\mathcal{C}$ obeys the Tambara-Yamagami fusion rules, then its associators are automatically continuous in the sense that $\mathcal{C}$ is obtained from a continuous tensor category by forgetting its topology.