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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).

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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. –  Narutaka OZAWA Feb 6 at 8:42
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@NarutakaOZAWA, but there is enough space to put that proof in the answer box below –  no identity May 6 at 8:17

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