The minimum norm of certain completely positive maps
Authors: Suen Ching Yun 
Published: July 2010

Let L be a completely bounded linear map from a unital

C
∗

{C^ \ast }

-algebra to the algebra of all bounded linear operators on a Hilbert space. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="min left-brace right-brace colon double-vertical-bar double-vertical-bar phi cb colon times times left-parenthesis right-parenthesis phi semicolon times times times amp semicolon semicolon times times times amp semicolon upper L semicolon times times times amp semicolon 0 semicolon times times times amp semicolon midline-horizontal-ellipsis semicolon times times times amp semicolon 0 semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon vertical-ellipsis upper L asterisk semicolon times times times amp semicolon semicolon times times times amp semicolon phi semicolon times times times amp semicolon upper L semicolon times times times amp semicolon semicolon times times times amp semicolon 00 semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon down-right-diagonal-ellipsis semicolon times times times amp semicolon semicolon times times times amp semicolon vertical-ellipsis semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon semicolon times times times amp semicolon upper L 0 semicolon times times times amp semicolon midline-horizontal-ellipsis semicolon times times times amp semicolon 0 semicolon times times times amp semicolon semicolon times times times amp semicolon upper L asterisk semicolon times times times amp semicolon phi times times nnis completelypositive separator for alln equals 2 upper S left-parenthesis upper L right-parenthesis comma">

min

{

‖
ϕ
‖

cb

:

(

ϕ

a
m
p
;

a
m
p
;
L

a
m
p
;
0

a
m
p
;
⋯

a
m
p
;
0

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
⋮

L
∗

a
m
p
;

a
m
p
;
ϕ

a
m
p
;
L

a
m
p
;

a
m
p
;
0

0

a
m
p
;

a
m
p
;

a
m
p
;
⋱

a
m
p
;

a
m
p
;

⋮

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
L

0

a
m
p
;
⋯

a
m
p
;
0

a
m
p
;

a
m
p
;

L
∗

a
m
p
;
ϕ

)

n
×
n

is completely

positive

for all

n

}

=
2
S
(
L
)
,

\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
{

‖
ϕ
‖

cb

:
ϕ
±
Re
⁡
λ
L

S(L) = \min \{ {\left \| \phi \right \|_{{\text {cb}}}}:\phi \pm \operatorname {Re} \lambda L

is completely positive for all

|

λ

|

=
1
}

|\lambda | = 1\}

.