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Fullness and Connes’ $$\tau$$ invariant of type III tensor product factors. (English. French summary) Zbl 1417.46042
Summary: We show that the tensor product $$M\overline{\otimes} N$$ of any two full factors $$M$$ and $$N$$ (possibly of type III) is full and we compute Connes’ invariant $$\tau(M \overline{\otimes} N)$$ in terms of $$\tau(M)$$ and $$\tau(N)$$. The key novelty is an enhanced spectral gap property for full factors of type III. Moreover, for full factors of type III 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 I amenable factor $$P$$, the tensor product factor $$M \overline{\otimes} P$$ has a unique McDuff decomposition, up to stable unitary conjugacy.

##### MSC:
 46L36 Classification of factors 46L10 General theory of von Neumann algebras 46L40 Automorphisms of selfadjoint operator algebras 46L55 Noncommutative dynamical systems 46L06 Tensor products of $$C^*$$-algebras
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