Probably isomorphic structures

TL;DR

This paper introduces the concept of probably isomorphic structures, showing their equivalence to $L^1$-space isomorphisms in specific cases, and applies set-theoretic methods to von Neumann algebras.

Contribution

It characterizes probably isomorphic structures in terms of $L^1$-space isomorphisms and extends set-theoretic techniques to von Neumann algebra tensorial primeness.

Findings

Probably isomorphic structures are characterized by $L^1$-space isomorphisms in certain classes.

Set-theoretic arguments are used to analyze ultraproducts of von Neumann algebras.

Nontrivial ultraproducts of diffuse von Neumann algebras are tensorially prime under specific conditions.

Abstract

Two structures $M, N$ in the same language are called probably isomorphic if they (or, in case of metric structures, their completions) are isomorphic after forcing with the Lebesgue measure algebra. We show that, if $M$ and $N$ are discrete structures, or extremal models of a non-degenerate simplicial theory, then $M$ and $N$ are probably isomorphic if and only if $L^1([0,1], M) \cong L^1([0,1], N)$. We moreover employ some of the set-theoretic arguments used to prove the aforementioned result to characterize when nontrivial ultraproducts of diffuse von Neumann algebras are tensorially prime.