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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