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

Abstract

We show that the tensor product $M\overline{\otimes }N$ of any two full factors $M$ and $N$ (possibly of type $\mathrm{III}$) 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 $\mathrm{III}$. Moreover, for full factors of type $\mathrm{III}$ 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 $\mathrm{I}$ amenable factor $P$, the tensor product factor $M\overline{\otimes }P$ has a unique McDuff decomposition, up to stable unitary conjugacy.

22 pages. v3: section on generalized Bernoulli crossed products and appendix removed

References in corpus (2)

• Classification of type III Bernoulli crossed products
• Amenable absorption in amalgamated free product von Neumann algebras

Cited by in corpus (4)

• Structure of bicentralizer algebras and inclusions of type III factors
• Structure of extensions of free Araki-Woods factors
• Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras
• Pointwise inner automorphisms of almost periodic factors