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

TL;DR

This paper proves that the tensor product of any two full factors, including type III, remains full and provides a formula for Connes' τ invariant, with applications to uniqueness of McDuff decompositions.

Contribution

It introduces an enhanced spectral gap property for full type III factors and computes the τ invariant for their tensor products, extending previous results.

Findings

Tensor product of full factors is full.

Explicit computation of τ invariant for tensor products.

Uniqueness of McDuff decomposition for certain tensor products.

Abstract

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