A Note on Numerical Radius Operator Spaces

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http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.76085 Jun. 19, 2019 1251 Journal of Applied Mathematics and Physics

A Note on Numerical Radius Operator Spaces

Yuanyi Wang1_{, Yafei Zhao}2

1_{College of Science and Technology, Ningbo University, Ningbo, China}

2_{Department of Mathematics, Zhejiang International Studies University, Hangzhou, China}

Abstract

In this paper, we first study some -completely bounded maps between various numerical radius operator spaces. We also study the dual space of a numerical radius operator space and show that it has a dual realization. At last, we define two special numerical radius operator spaces MinE and

MaxE which can be seen as a quantization of norm space E.

Keywords

Numerical Radius Operator Space, Dual Space, Quantization

1. Introduction and Preliminaries

The theory of operator space is a recently arising area in modern analysis, which is a natural non-commutative quantization of Banach space theory. An operator space is a norm closed subspace of  

( )

. The study of operator space begins with Arverson’s [1] discovery of an analogue of the Hahn-Banach theorem. Since the discovery of an abstract characterization of operator space by Ruan [2], there have been many more applications of operator space to other branches in functional analysis. Effros and Ruan studied the mapping spaces CB V W

(

,

)

in [3] and the minimal and maximal operator spaces in [4]. The fundamental and systematic developments in the theory of tensor product of operator spaces can be found in [5] [6]. The tensor products provide a fruitful approach to mapping spaces and local property. For example, Effros, Ozawa and Ruan [7] showed that an operator space V is nuclear if and only if V is locally reflexive and **

V is

in-jective. Dong and Ruan [8] showed that an operator space V is exact if and only if V is locally reflexive and **

V is weak* exact. In [9], Han showed that an

op-erator space V satisfies condition C if and only if it satisfies conditions C′_and C′′_{. Based on the work of Han, Wang}_[10]_{gave a characterization of condition}

C∧′ on the operator spaces. Amini, Medghalchi and Nikpey [11] proved that an

How to cite this paper: Wang, Y.Y. and Zhao, Y.F. (2019) A Note on Numerical Radius Operator Spaces. Journal of Applied Mathematics and Physics, 7, 1251-1262.

https://doi.org/10.4236/jamp.2019.76085

Received: May 14, 2019 Accepted: June 16, 2019 Published: June 19, 2019

http://creativecommons.org/licenses/by/4.0/

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DOI:10.4236/jamp.2019.76085 1252 Journal of Applied Mathematics and Physics injective operator space is global exactness if and only if it is reflexive. The read-ers may refer to [12] [13] for the basics on operator spaces.

Recently, some new algebraic structures derived from operator spaces also have been intensively studied. An operator system is a matrix ordered operator space which plays a profound role in mathematical physics. Kavruk, Paulsen, Todorov and Tomforde gave a systematic study of tensor products and local property of operator systems in [14] [15]. In [16], Luthra and Kumar showed that an operator system is exact if and only if it can be embedded into a Cuntz algebra. The numerical radius operator space is also an important algebraic structure which is introduced by Itoh and Nagisa [17] [18]. The conditions to be a numerical radius space are weaker than the Ruan’s axiom for an operator space. It is shown that there is a -complete isometry from a numerical radius oper-ator space into a Hilbert space with numerical radius norm. They also studied many relations between the operator spaces and the numerical radius operator spaces. The category of operator space can be regarded as a subcategory of nu-merical radius operator space.

We now recall some concepts needed in our paper. An (abstract) operator space is a complex linear space V together with a sequence of norms n

( )

⋅ on

the n n× matrix space M_n

( )

V for each n∈, which satisfies the following Ruan’s axioms OI, OII:

( )

{

}

0

OI : max , ;

0

m n m n

v

v w

w

+

 

=

 

 

  

(

)

( )

OII :_n α βv ≤ α _m v β

for all v∈Mm

( )

V ,w∈Mn

( )

V and α∈Mn m,

( )

 ,β∈Mm n,

( )

 . If V is an (ab-stract) operator space, then there is a complete isometry Ψ from V to  

( )

, that is,

( )

i j, n

(

i j,

)

n

v v

Ψ  = _ _

   for all vi j,  ∈Mn

( )

V ,n∈.

An abstract numerical radius operator space is a complex linear space V to-gether with a sequence of norms n

( )

⋅ on the n n× matrix space Mn

( )

V

for each n∈, which satisfies the following axioms WI, WII:

( )

{

}

0

WI : max , ;

0

m n m n

v

v w

w

+

 

=

 

 

  

(

)

2

( )

WII :n α αv ≤ α m v

for all v∈Mm

( )

V ,w∈Mn

( )

W and α∈Mn m,

( )

 . Let ω

( )

⋅ be the numerical radius norm on  

( )

. If V is an abstract numerical radius operator space, then there is a -complete isometry Φ from

(

V,n

)

to

(

 

( )

,ωn

)

, that is,

( )

(

,

)

(

,

)

n vi j n vi j

ω Φ = _ _ for all vi j,  ∈ Mn

( )

V ,n∈. Given a numerical

ra-dius operator

(

V,n

)

, we can define an operator space

(

V,n

)

by

( )

2

0 1

:

0 0

2 n n

v OW v = __ __

 

 

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DOI:10.4236/jamp.2019.76085 1253 Journal of Applied Mathematics and Physics Given abstract numerical radius operator spaces (or operator spaces) V W,

and a linear map ϕ from V to W, ϕn from Mn

( )

V to Mn

( )

W is defined

to be ϕn

(

vi j, 

)

for each vi j,  ∈Mn

( )

V ,n∈. We use a simple notation for

the norm of v=vi j, ∈Mn

( )

V to be 

( )

v (resp. 

( )

v ) instead of n

( )

v

(resp. n

( )

v ), and for the norm of

( )

*

n

f∈M V to be

( )

{

( )

}

*

,

sup : _{i j} _n , 1

f = f v v=_v _∈M V v ≤

  .

We denote the norm ϕn by

( )

ϕn =sup

{

(

ϕn

( )

v

)

:v=vi j, ∈Mn

( )

V ,

( )

v ≤1

}

  

(resp. 

( )

ϕn =sup

{



(

ϕn

( )

v

)

:x=vi j, ∈Mn

( ) ( )

V , v ≤1

}

). The -completely bounded norm (resp. completely bounded norm) of ϕ is de-fined to be

( )

sup

{

( )

_n :

}

cb n

ϕ = ϕ ∈

  , (resp.

( )

sup

{

( )

_n :

}

cb n

ϕ = ϕ ∈

  ).

We say ϕ is -completely bounded (resp. completely bounded) if

( )

ϕ _cb< ∞

 (resp. 

( )

ϕ cb< ∞), and ϕ is -completely contractive (resp.

completely contractive) if 

( )

ϕ _cb ≤1 (resp. 

( )

ϕ _cb ≤1). We call ϕ is a 

-complete isometry (resp. complete isometry) if 

(

ϕn

( )

v

)

=

( )

v (resp.

( )

(

ϕn v

)

=

( )

v

  ) for each x∈Mn

( )

V ,n∈.

In Section 2, we study the bounded maps on finite dimension numerical ra-dius operators and commutation C*-algebras. We prove these maps are all  -completely bounded. In Section 3, we study the dual space of a numerical ra-dius operator space and prove its dual space has a dual realization on a Hil-bert space . In Section 4, we define the numerical radius operator spaces

MinE and MaxE for a normed space E, and prove that

(

_MaxE

)

*₌_MinE* and _MaxE*₌

(

_MinE

)

*.

In order to improve the readability of the paper, we give an index of notation:

Index of Notation ( )

  Hilbert space

( ) ,

m n

M V m by n matrix space over V

( ) 0

C Ω Space of continuous complex functions vanishing at ∞ on Ω

α Matrix norm of α

,

v w Scalar pairing of matrices

,

v w Matrix pairing of matrices

(V,n) Operator space

(V,n) Numerical radius operator space ( )

ω ⋅ Numerical radius norm on  ( )

( , )

B V W Space of bounded mappings

( , )

Bσ V W Space of w*-bounded mappings

( , )

CB V W Space of completely bounded mappings

( , ) CB V W

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DOI:10.4236/jamp.2019.76085 1254 Journal of Applied Mathematics and Physics

2. Bound Linear Maps

In this section, we study some bounded linear maps on the numerical radius op-erator spaces.

Proposition 2.1. If

(

V,_n

)

is an operator space and

(

V,_n

)

is a numerical

radius operator space satisfies v =1, then the mapping

: :

v C V v

θ → α→α

is -completely isometric.

Proof. Since max

( )

 =ω

( )

 , by Lemma 3.8 and 3.9 in [18], we have

( )

(

θv: max  → v

)

cb ≤

( )

θv cb =1

   

and

( )

_v

(

_v:

( )

)

.

cb v cb

θ ≤ θ ω  →

  

So

( )

(

_v:

)

( )

_v 1.

cb cb

v

θ ω  → = θ =

   

Now we consider the condition for finite dimensional numerical radius oper-ator spaces.

Proposition 2.2. Given abstract operator spaces

(

V,_n

)

and

(

W,_n

)

with

either V or W n-dimensional,

(

V,_n

)

and

(

W,_n

)

are numerical radius

oper-ator spaces, any linear mapping ϕ:V →W satisfies

( )

(

:

)

(

:

( )

)

.

cb

V W n V W

ϕ → ≤ ϕ →

     

Proof. Let us suppose that W has dimension n. We may select an Auerbach basis for W, which by definition is a vector basis w w1, 2,,wn with 

( )

wj =1, there

exist

( )

*

j

g ∈ W with 

( )

g_j =1 and gj

( )

wi =δij. Since

1

.

j

n

W w j j

id θ g

=

∑



We have

1

,

j

n w j j

g

ϕ θ ϕ

=

∑

_{ }

where θw_j

( )

α =αwj are -complete isometries from  to W, and gjϕ

are bounded linear functionals on V. It follows from Lemma 2.3 in [18] that

( )

(

)

( )

(

)

(

( )

)

( )

(

)

( )

(

)

1

:

: :

:

: .

j

cb n

w j _cb

cb j

n

j j

V W

W g V

g V

n V W

ϕ

θ ω ϕ ω

ϕ ω

ϕ

=

→

≤ → ⋅ →

= →

≤ →

∑



 



  

   

 

  

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DOI:10.4236/jamp.2019.76085 1255 Journal of Applied Mathematics and Physics Proposition 2.3. If

(

V,n

)

and

(

W,n

)

are n-dimensional operator spaces,

(

V,_n

)

,

(

W,_n

)

are numerical radius operator spaces, then there exists a

li-near isomorphism ϕ:

( )

V →

( )

W such that

( )

(

)

(

1

( )

)

2

: : .

cb cb

V W W V n

ϕ _→ _⋅ ϕ− _→ _≤

     

Proof. We choose Auervach bases vi∈V and wi∈W i 1,

(

= ,n

)

, together

with dual bases

( )

*

i

f ∈ V and g_i∈

( )

W * with 

( )

f_i =

( )

g_i =1. We

have that

( )

1

: :

n

i i i

V W v f v w

ϕ

=

→ 

∑

and

( )

1

: :

n

i i i

W V w g w v

ψ

=

→ 

∑

are inverse linear mappings. Since

( )

(

)

( )

(

)

(

( )

)

1

:

: :

i

cb n

i _cb w _cb

i

V W

f V W n

ϕ

ω θ ω

=

→

≤

∑

→ _ ⋅ _ → ≤

  

   

and similarly

( )

(

:

)

,

cb

W V n

ψ → ≤

  

the result follows.  For any commutative C*-algebra, we can assume that  coincides with

( )

0

C Ω . We may identify M_n

(

C₀

( )

Ω

)

with C0

(

Ω,Mn

)

. When given

( )

(

0

)

ij n

f = _{ }f ∈M C Ω , we define

( )

sup

{

(

ij

( )

)

}

,

w

f ω f w

∈Ω

=



then C0

( )

Ω can be seen as a numerical radius operator space. We call such  a commutative C*-algebra with a numerical radius norm.

Theorem 2.4. Let V be a numerical radius operator space, and let  be a commutative C*-algebra with a numerical radius norm. Then any bounded li-near mapping ϕ:V → satisfies 

( )

ϕ _cb =

( )

ϕ .

Proof. We can assume that  coincides with C0

( )

Ω . Taking the supre-mum over all w∈ Ω and α ∈n_with

2 1

α = , we have

( )

(

)

(

( )

)

{

(

( )

)

}

( )

{

}

{

( )

}

, ,

sup

sup sup

n i j i j

w

i j i i j j

w w

v v v w

v w v w

α α

ϕ ϕ ω ϕ

ϕ α α α ϕ α

∈Ω

= =

∑

 

and thus letting α also stand for column matrices,

( )

(

)

{

(

*

)

( )

}

{

(

*

)

}

( ) ( )

,

sup ( ) sup .

n

w

v v w v v

α α

ϕ = ϕ α α ≤ ϕ α α ≤ ϕ

    

This shows that that 

( )

ϕn ≤

( )

ϕ for all n∈, and thus

( )

ϕ cb =

( )

ϕ

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3. Dual Spaces of Numerical Radius Operator Spaces

In this section, we introduce a lemma first.

Lemma 3.1. Suppose that V is a numerical radius operator space. Given any element v∈Mn

( )

V , there exists a -complete contraction ϕ:V→Mn such that ω ϕ

(

n

( )

v

)

=

( )

v .

Proof. If we are given v∈Mn

( )

V , then we may use the Hahn-Banach theorem to

find a linear functional

( )

*

n

F∈ _M V _ with *

( )

F =1 and F v

( )

=

( )

v . From Lemma 2.4 in [18], there is a corresponding -complete contraction

:V Mn

ϕ → _{for which}

( )

(

_n v

)

_n

( )

v , F v

( )

v .

ω ϕ ≥ ϕ ξ ξ = =

The reverse inequality is trivial.  There is a natural numerical radius operator space structure on the mapping space CB V W

(

,

)

. In this paper, we consider the dual space

(

)

(

)

*

, ,

V =B V  =CB V  .

Our task is to define

( )

*

n

M V by introducing an appropriate norm on M_n

( )

V* .

Each

( )

*

,

i j n

f =_f _∈M V determines a linear mapping f V: →Mn, where

( )

i j,

( )

f v = f v _. This gives us a linear isomorphism M_n

( )

V* ≅CB V M

(

, _n

)

,

which we use to determine the norm on

( )

*

n

M V . Thus, if we let

( )

*

n

M V be

the corresponding normed space, we have the isometric identification

( )

*

(

_,

)

n n

M V =CB V M .

For any

( )

*

n

f∈M V , we have from Lemma 2.3 in [18],

( )

{

(

( )

)

( )

}

(

)

( )

{

}

sup : , 1

sup , : , 1 ,

n n

n

f f v v M V v

f v v M V v

ω ω

= ∈ ≤

 



where ⋅ ⋅, is the matrix pairing. Conversely, the norm on

( )

*

n

M V deter-mines that on Mn

( )

V . Since we have from Lemma 3.1 that for any v∈Mn

( )

V ,

( )

{

(

( )

)

(

)

( )

}

(

)

(

)

( )

{

}

sup : , , 1

sup , : , , 1 .

n n cb

n cb

v f v f CB V M f

f v f CB V M f

ω ω

= ∈ ≤

  

 

Proposition 3.2. The matrix norms on *

V determine a numerical radius

operator space.

Proof. Let us suppose that we are given

( )

*

n

f ∈M V , α∈Mn m, . Then

(

)

(

)

(

) (

)

( )

* *

2

,

r r r r

r r

cb

f I f I

I f

f

α α α α

α α

= ⊗ ⊗

≤ ⊗

≤

 

and hence

(

*

)

2

( )

cb cb

f f

α α ≤ α

  . We have WII.

On the other hand, given

( )

*

( )

*

,

n n

f ∈M V g∈M V , and v∈Mr

( )

V with

( )

v ≤1

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(

) ( )

(

)

(

( ) ( )

)

( )

(

)

( )

(

)

(

( )

)

{

}

( )

{

}

, , ,

max ,

max , ,

i j i j r

r r i j r r

cb cb

f g v f v g v

f v g v f v g v

f g

ω ω

ω

ω ω

 

⊕ = _ ⊕ _

= ⊕

=

≤  

and hence

(

)

max

{

( )

,

( )

}

cb cb cb

f ⊕g ≤ f g

   . We have WI. 

If ϕ:V →W is -completely bounded mapping, then we let ϕ*_:_W*_→_V* be the dual Banach space mapping. For any v∈Mn

( )

V and

( )

*

m

g∈M W , we have

( )

(

( )

)

*

( )

*

( )

, , , ,

, _n _{k l} _{i j} _{k l} _{i j} , .

m

gϕ v =_g ϕ v  _=_ϕ g v _= ϕ g v

Proposition 3.3. Given numerical radius operator spaces V and W, and a  -completely bounded mapping ϕ:V →W, we have

( )

*

( )

n n

ϕ = ϕ

  for

all n∈, and

( )

*

( )

cb cb

ϕ = ϕ

  .

Proof. The second relation is immediate from the first. The first follows from the calculation

( )

*

{

( )

*

( )

}

{

(

( )

)

}

( )

sup , sup , _n _n ,

n n g v g v

ϕ = ω ϕ = ϕ = ϕ

  

where the supermum is taken over all

( )

*

n

g∈M W and v∈Mn

( )

v of norm

less than 1.  We also note that given a -completely bounded mapping ϕ∈CB V W

(

,

)

,

its second adjoint mapping ϕ**_:_V** →_W** is in

(

** **

)

,

CB V W

 with

( )

**

( )

cb cb

ϕ = ϕ

  , where ϕ** restricted to V is equal to ϕ_.

Given a numerical radius operator space W which is the dual of a complete numerical radius operator space V, and a Hilbert space , we say that a map-ping ϕ:W →

(

 

( )

,ω

)

is a dual realization of W on , if it is a weak* ho-meomorphic -completely isometric injection.

Theorem 3.4. If V is a complete numerical radius operator space, then *

V

has a dual realization on a Hilbert space . Proof. Let sn Mn

( )

V _{( )}1

σ

⋅ ≤

= _ . We have from Lemma 2.3 [18] that if

( )

*

(

)

,

n n

f∈M V =CB V M , then 

( )

f =sup

{

ω

(

ϕ,f

)

,ϕ∈sσ_n

}

. We de-fine s n sn

σ σ

∈

=

_

_ and we let n( )

sσ

ϕ ϕ∈

= ⊕ _

 , where n=n

( )

ϕ is the integer with sn

σ

ϕ∈ . The argument in the proof of Theorem 2.1 in [18] shows that the mapping

( )

(

( )

)

*

: :

s

V f ϕ f _ϕ_∈σ

Φ →  →

is a -complete isometry. It is obvious that the mapping Φ_{is continuous in} the weak* topology. Since V_{( )}⋅ ≤1 is weak* compact, then its domain

(

( )

)

* 1 V _{⋅ ≤}

Φ _

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DOI:10.4236/jamp.2019.76085 1258 Journal of Applied Mathematics and Physics one-to-one and weak* continuous on *_{( )}

1

V__{⋅ ≤} , thus it is a weak* homeomor-phism. Since V is complete, Φ_maps *

V weak* homeomorphically onto its

image.  Proposition 3.5. If W is complete, then so is CB V W

(

,

)

.

Proof. Let us suppose that W is complete. It suffices to show that

(

,

)

CB V W

 is a closed subspace of B V W

(

,

)

. Given any Cauchy sequence

(

,

)

n CB V W

ϕ ∈ , it is clear that ϕn is a Cauchy sequence in B V W

(

,

)

. From classical Banach space theory, B V W

(

,

)

is complete, and thus there is a

bounded linear mapping ϕ:V →W such that ϕn converges to ϕ in the norm topology, i.e., 

(

ϕ_n−ϕ

)

→0. Since ϕ_n is Cauchy in CB V W

(

,

)

, for

any ε>0 there exist a sufficiently large integer N

( )

ε >0 such that whenever

( )

,

n m>N ε , we have

(

ϕ ϕ_n− _{m cb}

)

<ε.



Given any v=vi j, ∈Mp

( )

V and p∈N, we have

(

) ( )

(

_n _m

)

(

_n _m

)

( )

.

p v cb v v

ϕ ϕ− ≤ ϕ ϕ− ⋅ <ε

   

Since ϕm

( )

vi j, converges to ϕ

( )

vi j, in W, we have

(

) ( )

(

_n

)

( )

,

p v v

ϕ −ϕ ≤ε

 

and thus 

(

ϕ ϕn−

)

cb≤ε. It follows that ϕ∈CB V W

(

,

)

and ϕn converges to ϕ in CB V W

(

,

)

. 

4. The Min and Max Numerical Radius Operator Spaces

We let η denote the category of normed spaces, in which the objects are the normed spaces and the morphisms are the bounded linear mappings. Similarly, we let D be the category of numerical radius operator spaces with the

mor-phisms being the -completely bounded mappings. We have a natural “for-getful” functor N:D→η which maps a numerical radius into its underlying normed space. We say that a functor Q:η →D is a strict quantization if for each normed space E, N Q E

( )

=E, and for each bounded linear mapping of normed space ϕ:E→F, the corresponding mapping Q

( ) ( )

ϕ :Q E →Q F

( )

satisfies 

(

Q

( )

ϕ

)

_cb =

( )

ϕ .

For any Banach space E, we let b_r

( )

E =B E M

(

, _r

)

__{( )}_{⋅ ≤}₁ and

( )

r N r

( )

b E =



_∈ b E . We define the matrix norms

( )

Min

v

 and 

( )

v Max for

( )

n

v∈M E by

( )

sup

{

(

_n

( )

)

:

( )

}

Min

v = ω f v f∈b E 

and

( )

v Max=sup

{

ω

(

fn

( )

v

)

: f∈b E1

( )

}

. 

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oper-DOI:10.4236/jamp.2019.76085 1259 Journal of Applied Mathematics and Physics ator spaces.

Proof. To see that these are indeed numerical radius operator space matrix norms, it suffices to consider the linear injections

( )

(

1

)

:

(

( )

)

_{f b E}₁_{( )} E→_∞ b E V→ f v _∈ and

( )

(

,

)

:

(

( )

)

_{( )}

)

r

r r _{f b} _E

r r

E ∞ b E M V f v _∈

∈ ∈

→

∏

_ →

 

respectively. We have the natural numerical radius operator space identifications

( )

(

b E1

)

b E1( )

∞ =

∏

  and

∏

r∈∞

(

br

( )

E ,Mr

)

=

∏ ∏

r∈ br( )E Mr. 

Since the relative matrix norms on E are given above, it is evident that these determine numerical radius operator spaces, which we denote by Min E and Max E, respectively. We refer to these numerical radius operator spaces as the minimal and the maximal quantization of E.

If V is a numerical radius operator space and v∈Mn

( )

V , then it follows

from Lemma 2.3 in [18] that

( )

v =sup

{

ω

(

f_n

( )

v

)

: f∈s_n

( )

V

}

.



Since b v1

( )

⊆sn

( )

v ⊆b v

( )

, we conclude that 

( )

v Min ≤

( )

v ≤

( )

v Max

for any v∈Mn

( )

V .

Proposition 4.2. For any numerical radius operator space V and normed space E, and any linear mapping ϕ:V →E, we have

(

,

)

(

:

)

(

:

)

.

cb

CB V MinE = ϕ V →MinE = ϕ V →E

  

Proof. Let us suppose that v∈Mn

( )

V and 

( )

v ≤1. Then

( )

(

_n

)

sup

{

(

_n _n

( )

)

:

(

:

)

1 .

}

Min

v f v f E

ϕ = ϕ → ≤

  

But 

( )

f ≤1 implies that

(

)

(

f ϕ _n

)

≤

(

f ϕ

)

_cb =

(

f ϕ

)

≤

( )

f ⋅

( )

ϕ ≤

( )

ϕ

     

and thus 

(

ϕn

( )

v

)

_Min≤

( )

ϕ for all n∈. The inversion is clear.  If ϕ:E→F is a contraction, then since ϕ:MinE→MinF is a contraction,

= :

Minϕ ϕ MinE→MinF

is -completely contractive. We conclude that Min is a strict quantization functor. If ϕ:E→F is an isometric injection, then it follows that ϕ is 

-completely isometric since we may extend any f∈b E1

( )

to a functional

( )

1

f∈b F .

Proposition 4.3. For any normed space E and numerical radius operator space W, we have

(

,

)

(

,

)

,

CB MaxE W =B E W 

i.e., for any linear mapping ϕ:E→W,

(

:

)

(

:

)

.

cb

MaxE W E W

ϕ → = ϕ →

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DOI:10.4236/jamp.2019.76085 1260 Journal of Applied Mathematics and Physics Proof. To prove this, it suffices to show that if 

( )

ϕ ≤1, then 

( )

ϕ _cb ≤1.

For any n∈ and v∈M_n

(

MaxE

)

, we have

( )

(

)

( )

(

)

(

)

{

}

(

) ( )

(

)

(

)

{

}

( )

(

)

(

)

{

}

( )

sup : : 1,

.

n

n n r cb r n

n r

Max

v

f v f W M r

g v g E M r

v

ϕ

ϕ ϕ

= → ≤ ∈

≤ → ≤ ∈

=

 

  



 



From the above, we conclude that 

( )

ϕ _cb≤1. 

In particular, if we are given normed spaces E and F and a contraction

:E F

ϕ → , since ϕ:E→MaxF is a contraction, we have

:

Maxϕ MaxE→MaxF

is a -complete contraction. Thus Max is a strict quantization.

If there is a contraction ψ :F →E such that ψ ϕ = idE, then Maxϕ is  -completely isometric since MaxψMaxϕ =idMaxE. This is also the case for the canonical injection **

E E , since any contraction f :E→Mn automatically extends to the contraction **_: **

n

f E →M .

Proposition 4.4 If D is a subset of V*( )⋅ ≤1, and the absolutely convex hull

( )

co D is weak* dense in *_{( )} 1

V__{⋅ ≤} . Then for any v∈Mn

( )

V ,

( )

v Min =sup

{

ω

(

fn

( )

v

)

:f ∈D

}

.



Proof. Let us suppose that ω

(

fn

( )

v

)

≤1 for all f ∈D. If g∈

∑

kt fk k

where fk∈D and

∑

ktk ≤1, then

( )

(

n

)

k

( ) ( )

k _n k 1.

k k

g v t f v t

ω =ω_ _≤ ≤



∑



∑

For the absolutely convex hull co D

( )

is weak* dense in *_{( )} 1

V__{⋅ ≤} , given an arbitrary element g∈V*( )⋅ ≤1, we may find a net gβ ∈co D

( )

converging to g

in the weak* topology. Then gβ

( )

vi j, converges to g v

( )

i j, in the numerical radius norm topology. It follows that ω

(

gn

( )

v

)

≤1, and thus 

( )

v Min≤1. 

For any v∈Mn

( )

E , the linear mappings ϕv:f → f v

( )

i j,  are just the weak* linear mappings from *

E into

(

M_n,ω

)

, and thus we have the isometric

identification

(

)

(

*

)

,

n n

M MinE =Bσ E M .

Theorem 4.5. Suppose E is a normed space, then

(

_MaxE

)

*₌_MinE*. Proof. Given a normed space E, n∈ and a linear mapping f :E→Mn, the second adjoint **

f provides an extension of f to a weak* continuous map-ping from **

E to M_n. This provides us with a natural identification

(

_,

)

(

**_,

)

n n

B E M =Bσ E M . Thus, we have the isometries

(

)

(

*

)

(

_,

)

(

_,

)

(

*

)

_.

n n n n

M MaxE =CB MaxE M =B E M =M MinE

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corres-DOI:10.4236/jamp.2019.76085 1261 Journal of Applied Mathematics and Physics ponding commutative C*-algebra, then we have a natural mapping

( )( ) ( )

*

: Z : x v v x .

δ Ω → δ =

It is a simple consequence of the bipolar theorem that co

(

δ

( )

Ω

)

is weak* dense in *_{( )}

1

Z__{⋅ ≤} . From our preceding observation, if a=   a_ij is an element of

( )

n

M Z , we have

( )

sup

{

(

( )

_ij

)

:

( )

}

sup

{

(

_ij

( )

)

:

}

,

Min

a = ω _f a _ f∈ Ω =δ ω _a x _ x∈ Ω



i.e., Mn

( )

Z ≅C0

(

Ω,Mn

)

. We conclude that as a numerical radius operator

space, Z is just the minimal quantization of its underlying Banach space, i.e., Z =MinZ.

Theorem 4.6. Suppose E is a normed space, then

(

)

* * MinE =MaxE . Proof. Given a normed space E, and an isometric injection E Z, where Z

is a commutative C*-algebra. We have a corresponding commutative diagram

** **

E Z

→

↓ ↓

→

where the first column is an isometry, the second column is a -complete isometry, and both rows are isometric. Since **

Z is a numerical radius operator

space, it determines the minimal numerical radius operator space structure on **

E , hence

(

_MinE

)

**₌_MinE**. Thus, we have the _-complete isometries

(

)

*

(

)

_**

* **

,

MaxE =MinE = MinE

and since these identifications are compatible with the dualities, we have the 

-complete isometry *

(

)

*

MaxE = MinE . 

5. Conclusion

In this paper, we study the bounded linear operators and the dual spaces of the numerical radius operator spaces. We found that many of the basic results about the numerical radius operator space can be inspired by the theory of op-erator space. In the future, we will study the tensor product theory and local property in the category of numerical radius operator spaces. We believe that the further developments of the numerical radius operator space theory could play an import role in the operator space theory as well as have its own intrin-sic merit.

Supported

Project partially supported by the National Natural Science Foundation of China (No. 11701301).

Conflicts of Interest

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References

[1] Arveson, W. (1969) Subalgebras of C-Algebras. Acta Mathematica, 123, 141-224.

https://doi.org/10.1007/BF02392388

[2] Ruan, Z.-J. (1988) Subspaces of C-Algebras. Journal of Functional Analysis, 76, 217-230. https://doi.org/10.1016/0022-1236(88)90057-2

[3] Effros, E.G. and Ruan, Z.-J. (1988) Representations of Operator Bimodules and Their Applications. Journal of Operator Theory, 19, 137-157.

[4] Effros, E.G. and Ruan, Z.-J. (1988) On Matricially Normed Spaces. Pacific Journal

of Mathematics, 132, 243-264. https://doi.org/10.2140/pjm.1988.132.243

[5] Blecher, D.P. and Paulsen, V.I. (1991) Tensor Products of Operator Spaces. Journal

of Functional Analysis, 99, 262-292. https://doi.org/10.1016/0022-1236(91)90042-4

[6] Effros, E.G. and Ruan, Z.-J. (1991) A New Approach to Operator Spaces. Canadian

Mathematical Bulletin, 34, 329-337. https://doi.org/10.4153/CMB-1991-053-x

[7] Effors, E.G., Ozawa, N. and Ruan, Z.-J. (2001) On Injectivity and Nuclearity for Operator Spaces. Duke Mathematical Journal, 110, 489-521.

https://doi.org/10.1215/S0012-7094-01-11032-6

[8] Dong, Z. and Ruan, Z.-J. (2007) Weak Exactness for Dual Operator Spaces. Journal

of Functional Analysis, 253, 373-397. https://doi.org/10.1016/j.jfa.2007.06.003

[9] Han, K.H. (2007) An Operator Space Approach to Condition C. Journal of

Mathe-matical Analysis and Applications, 336, 569-576.

https://doi.org/10.1016/j.jmaa.2007.02.074

[10] Wang, Y.Y. (2016) Condition C∧′ of Operator Spaces. Canadian Mathematical

Bulletin, 60, 1-7. https://doi.org/10.4153/CMB-2016-064-3

[11] Amini, M., Medghalchi, A. and Nikpey, H. (2018) Globally Exact Operator Spaces.

Glasnik Matematički, 53, 179-186. https://doi.org/10.3336/gm.53.1.12

[12] Effros, E.G. and Ruan, Z.-J. (2000) Operator Spaces, London Mathematical Society Monographs. New Series, Vol. 23, the Clarendon Press, Oxford University Press, New York.

[13] Pisier, G. (2003) Introduction to Operator Space Theory. London Mathematical So-ciety Lecture Note Series, Vol. 294. Cambridge University Press, Cambridge.

https://doi.org/10.1017/CBO9781107360235

[14] Kavruk, A., Paulsen, V., Todorov, I. and Tomforde, M. (2011) Tensor Produces of Operator Systems. Journal of Functional Analysis, 261, 267-299.

https://doi.org/10.1016/j.jfa.2011.03.014

[15] Kavruk, A., Paulsen, V., Todorov, I. and Tomforde, M. (2013) Quotients, Exactness, and Nuclearity in the Operator System Category. Advances in Mathematics, 235, 321-360.https://doi.org/10.1016/j.aim.2012.05.025

[16] Luthra, P. and Kumar, A. (2017) Embeddings and Envelopes of Exact Operator Systems. Bulletin of the Australian Mathematical Society, 96, 274-285.

https://doi.org/10.1017/S0004972717000284

[17] Itoh, T. and Nagisa, M. (2006) The Numerical Radius Haagerup Norm and Hilbert Space Square Factorizations. Journal of the Mathematical Society of Japan, 58, 363-377. https://doi.org/10.2969/jmsj/1149166780

[18] Itoh, T. and Nagisa, M. (2006) Numerical Radius Norms on Operator Space.

Jour-nal of the London Mathematical Society, 74, 154-166.