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Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I am interested in $\mathcal{B}(H)$ only as a Banach algebra (operator algebra).

Do there exist two infinite-dimensional Banach algebras $A, B$ such that $\mathcal{B}(H)$ is isomorphic as a Banach algebra to the projective tensor product $A\otimes_\gamma B$?

You may also replace the projective tensor product by any other Banach algebra tensor product which arises from a reasonable crossnorm (so the vNA tensor product is not good).

Improve this question
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    $\begingroup$ No. Even the Banach algebra structure is irrelevant here. ${\cal B}(H)$ (and its kin) is not Banach isomorphic to a reasonable tensor product of two infinite-dimensional Banach spaces. The proof involves a few deep known facts about Banach space structure of ${\cal B}(H)$ and this margin is too small to contain it. $\endgroup$
    – Narutaka OZAWA
    Commented Feb 6, 2014 at 8:42
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    $\begingroup$ @NarutakaOZAWA, but there is enough space to put that proof in the answer box below $\endgroup$
    – Norbert
    Commented May 6, 2014 at 8:17
  • $\begingroup$ @Norbert please see here arxiv.org/abs/1412.3621 $\endgroup$
    – Tomasz Kania
    Commented Jun 1, 2017 at 16:44
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    $\begingroup$ @TomekKania, you should post this as an answer! $\endgroup$
    – Norbert
    Commented Jun 2, 2017 at 8:38
  • $\begingroup$ @Norbert this does not answer the question as the paper deals only with C*-tensor norms. $\endgroup$
    – Tomasz Kania
    Commented Jun 2, 2017 at 8:48

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