Tight decomposition of factors and the single generation problem

A II$_1$ factor $M$ has the {\it stable single generation} ({\it 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 {\it tight} decomposition, i.e., there exists a pair of hyperfinite II$_1$ subfactors $R_0, R_1 \subset M$ such that $R_0 \vee R_1^{op}=\Cal B(L^2M)$. We explain why this conjecture is interesting and discuss possible approaches to prove it. We also prove some related results.

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