Structural properties of close II$_1$ factors

TL;DR

This paper investigates how various structural properties of II$_1$ factors, such as solidity and property $ ext{Gamma}$, are preserved under small perturbations and explores their behavior near tensor product factors.

Contribution

It demonstrates the transfer of key structural properties between close II$_1$ factors and analyzes their near tensor product structures.

Findings

Structural properties like solidity transfer between close II$_1$ factors.

Close II$_1$ factors near tensor products also decompose similarly.

Key properties such as strong solidity and Cartan masa uniqueness are preserved.

Abstract

We show that a number of key structural properties transfer between sufficiently close II$_1$ factors, including solidity, strong solidity, uniqueness of Cartan masas and property $\Gamma$. We also examine II$_1$ factors close to tensor product factors, showing that such factors also factorise as a tensor product in a fashion close to the original.