w completely bounded norms - hikaribounded linear operator on a hilbert space h. we extend the...
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International Journal of Mathematical Analysis
Vol. 11, 2017, no. 20, 987 - 998
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijma.2017.710130
W Completely Bounded Norms
Ching-Yun Suen
Foundational Sciences, Texas A&M University at Galveston
P.O. Box 1675, Galveston, Texas 77553-1675, USA
Copyright © 2017 Ching-Yun Suen. This article is distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
Let A be a unital *C -algebra and L be a linear map from A to the algebra of all
bounded linear operator on a Hilbert space H . We extend the theorem [7,
Theorem 2.7] as follows:
nxnILD
LD
ILD
LDI
2
*
2
*
2
)(00
..0
0.)(
00
:inf{
is completely positive
for all }n VTVTw (.):)(inf{ * is a minimal commutant representation with
isometry for }L , where 21 . We also generalize the inequality [9, Theorem
2.8] as follows:
cbwcbwLL
2121
and cbwcbw
LL12
21
,
for 21 21 .
Mathematics Subject Classification: 47A63
Keywords: operator radii, completely positive maps, completely bounded maps,
w completely bounded norms
988 Ching-Yun Suen
1. Introduction
Let nM denote the *C -algebra of complex nn matrices and )(HB the
algebra of all bounded linear operators on a Hilbert space H . Sz.-Nagy and C.
Foias introduced the class C . J. A. R. Holbrook defined the operator radii
)0(.)( w [3] by
}1
:0inf{)( CTr
rTw where )(HBT .
Let A and B be unital *C -algebras and let BAL : be a bounded linear map.
The map L is called positive if )(aL is positive whenever a is positive in A .
The map L is called completely positive ( c.p.)if nnn MBMAIL :
defined by
baLbaIL n )())(( is positive for all n .
We define ))(()( awaw where is an isometric *-isomorphism from A to
)(HB on a Hilbert space H , and we define the w norm of L by
)21(},1)(,:))((sup{
awAaaLwLw
.
The map L is said to be w completely bonded ( c.b.) if wnn ILsup is finite.
The c.b. w norm of L is defined by
wnncbwILL sup , )21( .
We shall use the notations cbwcb
LL1
and nn ILL . In [9, Proposition 2.4]
we proved that
cbwADcbw
LL2
2
where
10
)2(0D .
From [5 , Theorem 2.2], we know that every c.b. linear map from a unital *C -
algebra A to )(HB has a minimal commutant representation with isometry
( m.c.r.i. ) VTV (.)* where is a *-representation from A to )(KB , V is an
isometry from H to K , and T is an operator in the commutant of ).(A From [5,
Theorem 2.10] and [8, Corollary 3.11 and Corollary 3.12], we have
VTVTLcbw
(.):min{ *
2
is a m.c.r.i. for
*:min{}
L
LL is c.p.} .
W completely bounded norms 989
In Section 2, we shall prove that
nnILD
LD
ILD
LDI
2
*
2
*
2
)(00
..0
0.)(
00
:inf{
is c.p. for all
VTVTwn (.):)(inf{} * is a m.c.r.i.for }L , where 21 .
When 1 , we have
nn
IL
L
IL
LI
2*
2*
2
0
0000
00
0..0
0.0
00
0000
0
:min{
is c.p.
for all }n VTVT (.):min{ * is a m.c.r.i. for cbw
LL2
} .
When 2 , we have [7, Theorem 2.7].
We know that )()(21 21 TwTw where 21 21 [1, Corollary 5]. In
Section 3, we shall prove inequalities involving w completely bounded norms as
follows:
cbwcbwLL
2121
and cbwcbw
LL12
21
, for 21 21 .
From the above inequalities, we have cbcbwcbw
LLL
22
,
2. Inequalities of w completely bounded norms
Proposition 2.1. Let L be a completely bounded map from a unital *C -algebra
A to the algebra )(HB of all bounded linear operators on a Hilbert space H . Let
LD and 2I be the maps from A to ))((2 HBM defined by
))(())(( aLDaLD where 21 and
)(0
0)())(( 2
a
aaI
,
respectively. Then
2
*
2
)(:min{
ILD
LDI
is c.p.}
cbwL
2
.
990 Ching-Yun Suen
Proof. Let VTV (.)* be a m.c.r.i. for L , then LD has a m.c.r.i. as follows:
V
V
a
aTD
V
V
aL
aLaLD
0
0
)(0
0)()(
0
0
)()1(0
)()2(0))((
*
.
The matrix
2
*
2
)( ITTD
TDIT
is positive, by [5, Proposition 2.6], we have
that the map
V
V
V
VTLD
LDV
V
V
VT
0
0
(.)0
0(.)
0
0)(
)(0
0
(.)0
0(.)
0
0
*
*
*
is c.p..
Now let VV (.)(.) * , we have
2
*
2
)(:min{
ILD
LDI
is c.p.
VTVT (.):min{} * is a m.c.r.i. for }L . Conversely,
let
2
*
2
)( ILD
LDI
be c.p., then
I
I
ILD
LDI
I
I
0
0
)(0
0
2
*
2
0)(
0
0
0*
2
2
LD
LD
I
I
for all 1 .
We have
LL
LILDI
Re)1(2)2(
)2(0)Re(
*22 is c.p., for all
1 .
Thus
L Re)1(2 is c.p., for all 1 .
Now let 0 where is a unital c.p. map with a minimal representation
VV (.)* .
Applying [5, Theorem 2.2], L has a m.c.r.i.. .(.)* VTV By [5, proposition 2.6],
W completely bounded norms 991
the matrix
2
*
2
)( ITD
TDI
is positive. Hence
VTVTTTD (.):min{ * is a m.c.r.i. for }L .0
By [5, Theorem 2.10] and [8, Corollary 3.12], we know that
VTVTLcbw
(.):min{ *
2
is a m.c.r.i. for }L and we prove the proposition.
Corollary 2.2.
*)(:min{
LD
LD is c.p. and ))((: 2 HBMA is
c.p.}cbwcbw
LL2
.
Proof. Let )(AMa n . Since )()(2
)( 22 awawaDw
,
we have
cbw
LD2
sup1n
}1)(:))(({ 22 awaLDw n cbw
ADL2
2
cbwcbwn
n
nn LLaLDw
aL
aLw
2
}1))((:)()1(0
)()2(0{sup 221
[8, Proposition 3.2].
From [8, Corollary 3.11], Proposition 2.1, and Corollary 2.2, we know that
cbw
LD2
*)(:min{
LD
LD is c.p. and ))((: 2 HBMA is
c.p. }
2
*
2
)(:min{
ILD
LDI
is c.p. }. We will discuss more general
case of nn matrices as follows:
Theorem 2.3. Let L be a completely bounded map from a unital *C -algebra A to
the algebra )(HB of all bounded linear operators on a Hilbert space H and the
map ))((: 22 HBMAI be defined by
)(0
0)())(( 2
a
aaI
. Then
nnILD
LD
ILD
LDI
2
*
2
*
2
)(00
..0
0.)(
00
:inf{
is c.p. for all }n
VTVTw (.):)(inf{ * is a m.c.r.i. for }L , where 21 .
992 Ching-Yun Suen
Proof. Let VTV (.)* be a m.c.r.i. for L , then LD has a m.c.r.i :
))(( aLD
V
V
a
aTD
V
V
0
0
)(0
0)()(
0
0*
.
From [7, Proposition 2.2], we know that
nnIkTD
TD
IkTD
TDIk
k
2
*
2
*
2
)(00
..0
0.)(
00
:min{
is positive for all }n
)()(2 2 TwTDw .
Let
V
V
V
V
0
0
(.)0
0(.)
0
0*
, then is unital and minimal.
By [5, Proposition 2.6], the map
nnTDwLD
LD
TDwLD
LDTDw
)(2)(00
..0
0.)(2)(
00)(2
2
*
2
*
2
is c.p..
Hence
nnILD
LD
ILD
LDI
2
*
2
*
2
)(00
..0
0.)(
00
:inf{
is c.p. for all }n
VTVTw (.):)(inf{ * is a m.c.r.i. for }L .
Conversely,
since
2
*
2
)( ILD
LDI
is positive, from the proof of Proposition 2.1, we
have VV (.)* and L has a m.c.r.i. VTV (.)* . By [5, Proposition 2.6],
the matrix
2
*
2
*
2
)(00
..0
0.)(
00
ITD
TD
ITD
TDI
is positive.
W completely bounded norms 993
Applying [7, Proposition 2.2], we have
).()(2 TwTDw
Hence
nnILD
LD
ILD
LDI
2
*
2
*
2
)(00
..0
0.)(
00
.:inf{
is c.p. for all }n
VTVTw (.):)(inf{ * is a m.c.r.i. for }L .
Corollary 2.4. Let L be a completely bounded map from a unital *C -algebra A
to the algebra )(HB of all bounded linear operators on a Hilbert space H and the
map ))((: 22 HBMAI be defined by
)(0
0)())(( 2
a
aaI
. Then
nnILD
LD
ILD
LDI
2
*
2
2
2
*
2
22
)(00
..0
0.)(
00
:inf{
is c.p. for all }n
VTVTw (.):)(inf{ * is a m.c.r.i. for }L , where 10 .
Proof. Applying the reciprocity law )()2()( 2 TwTw of Ando and
Nishio [1], We have the Corollary.
Corollary 2.5.
nn
IL
L
IL
LI
2*
2*
2
0
0000
00
0..0
0.0
00
0000
0
:min{
is c.p. for all }n
VTVT (.):min{ * is a m.c.r.i.for cbw
LL2
} .
Proof. Let 1 in Theorem 2.3. Applying [5, theorem 2.10] and [8, Corollary
3.11 and Corollary 3.12], we have the Corollary.
994 Ching-Yun Suen
Corollary 2.6.
nn
IL
L
IL
LI
2*
2*
2
0
0000
0
00..0
0.0
00
000
00
:min{
is c.p.for all
VTVTwn (.):)(min{2} *
2 is a m.c.r.i. for }L .
Proof. Let 2 in Theorem 2.3. By [6, Theorem 2.7], we extend [7, Theorem
2.7].
Example 2.7. Let )(: 2 CMCCL be defined by
0
0
2
1)(
b
abaL . Then from [6, Example 2.8] we have
,2,12
cbwcb
LL
and min VTVTw (.):)({ *
2 is a m.c.r.i. for 22
)21(}
L .
When 1 , we have
nn
IL
L
IL
LI
2*
2*
2
0
0000
00
0..0
0.0
00
0000
0
:min{
is c.p. for all 2} n .
When 2 , we have
W completely bounded norms 995
:min{
nn
IL
L
IL
LI
2*
2*
2
0
0000
0
00..0
0.0
00
000
00
is c.p. for all
2)21(} n .
3. Inequalities of w completely bounded norms
From [1, Corollary 5], we know that )(Tw is increasing when 21 . We
prove that cbw
L
is also increasing as follows:
Theorem 3.1. Let L be a completely bounded map from a unital *C -algebra A to
the algebra )(HB of all bounded linear operators on a Hilbert space H .Then
cbwcbwLL
2121
and cbwcbw
LL12
21
, where 21 21 .
Proof. Let )(AMa n . Since )()(12
awaw , we have
1
wnL sup }1)(:))(({11
awaLw n sup }1)(:))(({21
awaLw n
sup }1)(:))(({21
1
1 awa
Lw n
sup }1)(:))(({22
1
2 awa
Lw n
sup2
22
1
2
1
2 }1)(:))(({
wnn LawaLw .
Hence
cbwcbwLL
2121
.
Moreover,
2
wnL sup }1)(:))(({22
awaLw n sup })(:))(({ 22 22 awaLw n
sup })(:))(({ 21 12 awaLw n sup })(:))(({ 21 11
awaLw n
sup }1)(:))(({11
2
1 awaLw n
sup
111
1
2
2
1
2
1
1
2 }1)(:))(({
wnn LawaLw .
996 Ching-Yun Suen
Hence
.2112cbwcbw
LL
.
The following Corollary extends the inequality [9, Theorem 2.8]:
Corollary 3.2. cbcbwcbwcbw
LLLL
22 2
)1( , and
cbwcbwcbwLLL
22)1(
, where 21 .
Proof. Since n
nnnnnIwIDwDwDwDw
2))(
2())(())(()()1()( 2222
1
2
,
we have 2
1
. Applying Theorem 3.1, we have the Corollary.
From [10, Lemma 1], we know that cbwcbw
LL
2
, for 10 .
Corollary 3.3. cbwcbw
LL21
)2()2( 12
, and
cbwcbwLL
12
)2()2( 12
,
where 10 21 .
Proof. Since 2221 12 , by Theorem 3.1, we have the Corollary.
The inflated Schur product nnT MHBMSij
)(: is defined by
)())(( ijijijT TccSij
. [4].
Corollary 3.4. Let ))(()( HBMTU nij be a contraction, then n
ScbwU
2
,
where 21 .
Proof. By [8, Theorem 4.4 and Corollary 4.9], we have
nn
ji
V
V
V
TT
T
TT
V
V
V
TVVU
0.0
0...
..0
0.0
..
....
...
..
0.0
0...
..0
0.0
)(2
1
*
2
1
*, where T is a
contraction and iV is an isometry ),...,2,1( ni .
Then
W completely bounded norms 997
TnTU
1..1
....
....
1..1
.
By [8, Corollary 4.6], we know that n
ScbwU
1
2
.
By Theorem 3.1, we have n
SScbwUcbwU
22 2
.
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998 Ching-Yun Suen
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Received: October 20, 2017; Published: November 12, 2017
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