Fullness and Connes' $\mathit{\tau }$ invariant of type III tensor product factors

We show that the tensor product $MN$ of any two full factors $M$ and $N$ (possibly of type $\mathrm{I}\mathrm{I}\mathrm{I}$) is full and we compute Connes’ invariant $\tau (MN)$ in terms of $\tau (M)$ and $\tau (N)$. The key novelty is an enhanced spectral gap property for full factors of type $\mathrm{I}\mathrm{I}\mathrm{I}$. Moreover, for full factors of type $\mathrm{I}\mathrm{I}\mathrm{I}$ with almost periodic states, we prove an optimal spectral gap property. As an application of our main result, we also show that for any full factor $M$ and any non-type $\mathrm{I}$ amenable factor $P$, the tensor product factor $MP$ has a unique McDuff decomposition, up to stable unitary conjugacy.