# Journal of Operator Theory

Volume 85, Issue 1, Winter 2021 pp. 277-301.

Tight decomposition of factors and the single generation problem**Authors**: Sorin Popa

**Author institution:**Mathematics Department, University of California, Los Angeles, CA 90095-1555, U.S.A.

**Summary:**A II$_1$ factor $M$ has the \textit{stable single generation} (\textit{SSG}) property if any amplification $M^t$, $t>0$, 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 II$_1$ subfactors $R_0, R_1 \subset M$ such that $R_0 \vee R_1^\mathrm{op}=\mathcal B(L^2M)$. We provide supporting evidence, explain why the conjecture is interesting, and discuss possible approaches to settle it. We also prove some related results.

**DOI:**http://dx.doi.org/10.7900/jot.2019nov01.2279

**Keywords:**hyperfinite factor, coarse subfactors, coarse pairs, tight decomposition

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