Let $(x_n)$ be a sequence in a von Neumann algebra $M$ or its predual $M_*$. Is there a hyperfinite von Neumann subalgebra $N$ of $M$ such that $(x_n)\subset N$ or $N_*$?
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6$\begingroup$ Since every vN algebra acting on a separable Hilbert space is countably generated as a vN algebra, and every subfactor of a hyperfinite is hyperfinite, a set of generators (typically a set of two) for a nonhyperfinite factor (acting on a sep Hilbert space) provides a counter-example. $\endgroup$– David HandelmanCommented Nov 15, 2021 at 3:04
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