Tight decomposition of factors and the single generation problem

Abstract

A II1 factor M has the \textit{stable single generation} (\textit{SSG}) property if any amplification Mt, t>0, can be generated as a von Neumann algebra by a single element. We discuss a conjecture stating that if M is SSG, then M has a \textit{tight} decomposition, i.e., there exists a pair of hyperfinite II1 subfactors R0,R1⊂M such that R0∨Rop1=B(L2M). We provide supporting evidence, explain why the conjecture is interesting, and discuss possible approaches to settle it. We also prove some related results.