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Apr 28, 2016 at 15:43 comment added Yemon Choi Thanks for having a go at the question! I should note, writing $\otimes_\gamma$ for proj t.p. in category of Banach spaces, that "identity tensor transpose" is an isometry on from $M_n \otimes_\gamma M_n \to M_n \otimes_\gamma M_n$, because transpose is an isometric involution. Therefore, if $M_n \otimes_\gamma M_n$ coincides (up to equivalence of norms) with $M_n \widehat{\otimes} M_n$ then my problem is immediately solved...
Apr 27, 2016 at 16:47 history edited Mike Jury CC BY-SA 3.0
added remark pointing out gap in orginial argument
Apr 27, 2016 at 15:50 history answered Mike Jury CC BY-SA 3.0