# Tensor product decompositions and rigidity of full factors

## TL;DR

This paper establishes new rigidity results for tensor product decompositions of full factors, demonstrating limitations on their decompositions, stability of certain classes under tensor products, and providing prime factorization insights and counterexamples.

## Contribution

It introduces novel rigidity theorems for full factors, including countability of tensor decompositions, stability of classes under tensor products, and prime factorization results for crossed products.

## Key findings

- Full factors have at most countably many tensor product decompositions.
- Separable full factors with countable fundamental group are stable under tensor products.
- New primeness and unique prime factorization results for crossed products from higher rank lattices and Bernoulli shifts.

## Abstract

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.