Suppose $E$ is an operator space and $E^*$ is the dual operator space. It is well known that the matricial norm structure on $E^*$ is given by the formula $\|[f_{ij}]_{i,j=1}^n\|_{M_n(E^*)}:=\sup\{\|f_{ij}(v_{kl})\|:\|v\|_{M_n(E)}\leq 1\}.$ Is it true that if $[f_{ij}]_{i,j=1}^n$ is just a column matrix, then the supremum can be taken on the set where $v$ is a column matrix with norm $\leq 1$?
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asked Apr 24, 2021 at 10:19
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2$\begingroup$ You are asking in the wrong forum $\endgroup$– Gerald EdgarCommented Apr 24, 2021 at 14:51
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1$\begingroup$ @Gerald. Can you please elaborate what's wrong? $\endgroup$– A beginner mathmaticianCommented Apr 24, 2021 at 14:53
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$\begingroup$ If you read the information about mathovervflow you can find this: "MathOverflow's primary goal is for users to ask and answer mathematical questions related to current research in mathematics." $\endgroup$– Gerald EdgarCommented Apr 24, 2021 at 14:58
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3$\begingroup$ @GeraldEdgar I don't immediately see why either this topic or this question are not MO-worthy $\endgroup$– Yemon ChoiCommented Apr 24, 2021 at 17:45
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2$\begingroup$ I agree with @YemonChoi. That said, the question is not very well written: if we are meant to be considering a column vector then why write $[f_{ij}]_{i,j=1}^n$ with both $i$ and $j$ varying from $1$ to $n$? That's a matrix, not a column vector... Also, what evidence do you have for your conjecture: could you provide an example (non-trivial) Operator Space where it's true? $\endgroup$– Matthew DawsCommented Apr 24, 2021 at 20:43
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