## The minimum norm of certain completely positive maps

HTML articles powered by AMS MathViewer

- by Ching Yun Suen
- Proc. Amer. Math. Soc.
**123**(1995), 2407-2416 - DOI: https://doi.org/10.1090/S0002-9939-1995-1213870-2
- PDF | Request permission

## 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

- Uffe Haagerup and Pierre de la Harpe,
*The numerical radius of a nilpotent operator on a Hilbert space*, Proc. Amer. Math. Soc.**115**(1992), no. 2, 371–379. MR**1072339**, DOI 10.1090/S0002-9939-1992-1072339-6 - Vern I. Paulsen,
*Completely bounded maps and dilations*, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR**868472** - Vern I. Paulsen and Ching Yun Suen,
*Commutant representations of completely bounded maps*, J. Operator Theory**13**(1985), no. 1, 87–101. MR**768304** - C.-Y. Suen,
*The numerical radius of a completely bounded map*, Acta Math. Hungar.**59**(1992), no. 3-4, 283–289. MR**1171734**, DOI 10.1007/BF00050890

## 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