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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 $A\otimes_\gamma B$?

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  • $\begingroup$ Now cross-posted to MathOverflow: mathoverflow.net/questions/108188/tensorial-decomposition-of-bh $\endgroup$ – user16299 Sep 26 '12 at 22:09
  • $\begingroup$ I just now that the projective tensor product oh $H$ and $H^*$ is B(H). $\endgroup$ – Ali Bagheri Jan 23 '16 at 14:01
  • $\begingroup$ @AliBagheri Sorry, that's wrong: The completed projective tensor product $H\hat{\otimes}_\pi H$ is really "much smaller" than $\mathcal{B}(H)$ because it equals $\mathcal{l}^1(H)$, the Trace-class operators on $H$. $\endgroup$ – Hanno Mar 29 '17 at 12:47

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