Given two Banach algebras $A$ and $B$ with operator space structure on each of them, i.e both of them are closed subspaces of $B(H_1)$ and $B(H_2)$ respectively for some Hilbert spaces $H_1,H_2$. Does their operator space projective tensor product and Haagerup tensor products form Banach algebra? In other words is operator space projective/Haagerup tensor norm submultiplicative on $ A\otimes B$?
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2$\begingroup$ The answer is yes for the os-ptp if one assumes that the multiplication maps $A \times A \to A$ and $B\times B \to B$ are jointly completely bounded. My instinct is that if you do not make this assumption, then there will be counterexamples, but I cannot think of one immediately $\endgroup$– Yemon ChoiCommented Oct 31, 2019 at 15:25
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$\begingroup$ Any reference for your first statement? $\endgroup$– NewBCommented Oct 31, 2019 at 15:30
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1$\begingroup$ @OperatorZebra Look up the notion of a "completely bounded (or quantised) Banach algebra" e.g. in the book of Effros and Ruan. If $A\times A\rightarrow A$ is jcb then it extends to the os-ptp yielding a completely bounded map $A \widehat\otimes A\rightarrow A$. Then $(A\widehat\otimes B)\widehat\otimes(A\widehat\otimes B) \cong (A\widehat\otimes A)\widehat\otimes (B\widehat\otimes B) \rightarrow A \widehat\otimes B$ is completely bounded, so $A\widehat\otimes B$ is a (cb) Banach Algebra. $\endgroup$– Matthew DawsCommented Oct 31, 2019 at 15:54
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$\begingroup$ @MatthewDaws Thank you! That is exactly what I was looking for. Any words about Haagerup tensor product? $\endgroup$– NewBCommented Oct 31, 2019 at 21:53
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