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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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