oa.operator algebras - Does equality of the operator norm and the cb norm for every bimodule map over a C*-subalgebra imply that the subalgebra is matricially norming?

Dear Jonas,
My colleague Bill Johnson has drawn this to my attention.
Your question has a positive answer after noting that a $C$-bimodule map lifts to a $Cotimes M_n$-bimodule map on $Aotimes M_n$ If $X$ is an $ntimes n$ matrix over $A$ of norm 1 then for any row and column contractions $R$, $C$ over $C$ we have

$$|phi_n(X)|=sup{|Rphi_n(X)C|: |R|,|C|leq 1} =sup{|phi(RXC)|: |R|,|C|leq 1} leq |phi|$$
showing that $|phi_n|leq |phi|. Of course, the reverse inequality holds and we have proved that the norm and cb-norm coincide.
Best of luck with your studies,
Roger Smith

Sorry, I (mis)read your question rather quickly so the answer above is not an answer at all. Will think about it.

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