General solidity phenomena and anticoarse spaces for type $\mathrm{III}_1$ factors

TL;DR

This paper introduces anticoarse spaces in the context of type III factors, demonstrating that free Araki--Woods factors exhibit strong solidity and indecomposability properties using advanced entropy and random matrix techniques.

Contribution

It develops a new theory of anticoarse spaces for type III factors and shows free Araki--Woods factors possess unprecedented solidity and indecomposability features.

Findings

Free Araki--Woods factors satisfy strong solidity.

They exhibit the Peterson--Thom property.

The paper introduces anticoarse spaces in the infinite setting.

Abstract

By developing a theory of anticoarse spaces in the purely infinite setting and using 1-bounded entropy techniques along with recent strong convergence results in random matrix theory, we show that free Araki--Woods factors offer the first examples of type $\mathrm{III}$ factors satisfying vastly general degrees of indecomposability phenomena. Notably this includes strong solidity with respect to any weakening of the normalizer currently in the literature and the Peterson--Thom property.