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Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry
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Tensor product of von Neumann algebras
The answer to your question is No (beyond matrix cases i.e. finite dimensional von Neumann algebras). Even the von Neumann tensor product of one abelian von Neumann algebra, which is indeed $L^\infty( …