Does the following identity (bicontinuously) hold: $$\left(\mathbb{M}_n\bigotimes\nolimits_\epsilon X\right) \bigotimes\nolimits^\gamma \left( \mathbb{M}_m\bigotimes\nolimits_\epsilon Y\right)\cong \mathbb{M}_{nm}\bigotimes\nolimits_\epsilon\left(X\bigotimes\nolimits^\gamma Y\right),$$ where $X,Y$ are Banach spaces, $\bigotimes_\epsilon$ and $\bigotimes_\gamma$ denote respectively the Banach space injective and the Banach space projective tensor tensor products respectively. I am asking about the bicontinuity of the natural map given by $(a\otimes x)\otimes (b\otimes y)\mapsto (a\otimes b)\otimes (x\otimes y)$.

I think, one way to do it would be by identifying $\mathbb{M}_n\bigotimes\nolimits_\epsilon X$ (bicontinuously) with $\mathbb{M}_n\bigotimes\nolimits_\gamma X$ (and similar identification for $\mathbb{M}_n\bigotimes\nolimits_\epsilon Y$), and then use associativity and commutativity of the projective tensor product to get the conclusion. Is there another way to do it?

Will it still hold if the projective tensor product is replaced by some other Banach space cross tensor product, for which we have no idea about its commutativity or associativity?

Any reference would be highly appreciated.


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