Tensor product indecomposability results for existentially closed factors

TL;DR

This paper surveys deformation/rigidity theory results on tensor product decompositions of II$_1$ factors and introduces an uncountable family of existentially closed factors that resist such decompositions.

Contribution

It demonstrates the existence of uncountably many existentially closed II$_1$ factors that cannot be decomposed into tensor products with a full factor, advancing structural understanding.

Findings

Existence of uncountably many such factors.

These factors do not admit tensor decompositions with full factors.

Discussion of open problems in the structural theory of these factors.

Abstract

In the first part of the paper we survey several results from Popa's deformation/rigidity theory on the classification of tensor product decompositions of large natural classes of II$_1$ factors. Using a m\'elange of techniques from deformation/rigidity theory, model theory, and the recent works \cite{CIOS21,CDI22} we highlight an uncountable family of existentially closed II$_1$ factors $M$ which do not admit tensor product decompositions $M= P\bar \otimes Q$ into diffuse factors where $Q$ is full. In the last section we discuss several open problems regarding the structural theory of existentially closed factors.