Can we write any projection in the tensor product vN algebra $M\otimes N$ in terms of limits of projections $p\otimes q$, where $p$ and $q$ are projections in M, N or somewhat relate the projections onto their components!!
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2$\begingroup$ It can't be done for $M$ and $N$ finite dimensional factors (the rank of a pure tensor is the product of the ranks, and the unit ball is compact, ...), so why should we expect it to be true for infinite-dimensional ones? $\endgroup$– David HandelmanCommented Apr 23, 2019 at 8:46
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$\begingroup$ Expecting relations on the componenet projections $\endgroup$– user136400Commented Apr 23, 2019 at 9:06
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