A tensor product for representations of the Cuntz algebra and of the R. Thompson groups

The authors continue a series of articles studying certain unitary representations of the Richard Thompson groups $F,T,V$ called Pythagorean. They all extend to the Cuntz algebra $\mathcal{O}$ and conversely all representations of $\mathcal{O}$ are of this form. Via this approach we introduce a tensor product for a large class of representations of $F,T,V,\mathcal{O}$. We prove that a sub-category forms a tensor category and perform a number of explicit computations of fusion rules.