Internal sequential commutation and single generation

Abstract

We extract a precise internal description of the sequential commutation equivalence relation introduced in [KEP23] for tracial von Neumann algebras. As an application we prove that if a tracial von Neumann algebra $N$ is generated by unitaries $\{u_i{\}}_{i\in \mathbb{N}}$ such that $u_i\sim u_j$ (i.e, there exists a finite set of Haar unitaries $\{w_i{\}}_{i=1}^n$ in $N^{\mathcal{U}}$ such that $[u_i,w_1]=[w_k,w_{k+1}]=[w_n,u_j]=0$ for all $1\le k<n$) then $N$ is singly generated. This generalizes and recovers several known single generation phenomena for II${}_1$ factors in the literature with a unified proof.

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