The minimum norm of certain completely positive maps
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- by Ching Yun Suen
- Proc. Amer. Math. Soc. 123 (1995), 2407-2416
- DOI: https://doi.org/10.1090/S0002-9939-1995-1213870-2
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Abstract:
Let L be a completely bounded linear map from a unital ${C^ \ast }$-algebra to the algebra of all bounded linear operators on a Hilbert space. Then \[ \min \left \{ {{{\left \| \phi \right \|}_{{\text {cb}}}}:{{\left ( {\begin {array}{*{20}{c}} \phi \hfill & {} \hfill & L \hfill & 0 \hfill & \cdots \hfill & 0 \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & \vdots \hfill \\ {{L^ \ast }} \hfill & {} \hfill & \phi \hfill & L \hfill & {} \hfill & 0 \hfill \\ 0 \hfill & {} \hfill & {} \hfill & \ddots \hfill & {} \hfill & {} \hfill \\ \vdots \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & L \hfill \\ 0 \hfill & \cdots \hfill & 0 \hfill & {} \hfill & {{L^ \ast }} \hfill & \phi \hfill \\ \end {array} } \right )}_{n \times n}}\begin {array}{*{20}{c}} {{\text {is completely}}} \hfill \\ {{\text {positive}}} \hfill \\ {{\text {for all}}\;n} \hfill \\ \end {array} } \right \} = 2S(L),\] where $S(L) = \min \{ {\left \| \phi \right \|_{{\text {cb}}}}:\phi \pm \operatorname {Re} \lambda L$ is completely positive for all $|\lambda | = 1\}$.References
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Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2407-2416
- MSC: Primary 46L05; Secondary 47D25
- DOI: https://doi.org/10.1090/S0002-9939-1995-1213870-2
- MathSciNet review: 1213870