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Journal of Applied Mathematics and Physics, 2019, 7, 1251-1262

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 Wang

, Yafei Zhao

College of Science and Technology, Ningbo University, Ningbo, China

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

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

[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

is nuclear if and only if

is locally reflexive and

is in-

jective. Dong and Ruan [8] showed that an operator space

is exact if and only

is locally reflexive and

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

erator space

satisfies condition

if and only if it satisfies conditions

′

and

′′

. Based on the work of Han, Wang [10] gave a characterization of condition

∧

′

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

, 1251-1262.

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

Received:

May 14, 2019

Accepted:

June 16, 2019

Published:

June 19, 2019

9 by author(s) and

Scientific

Research Publishing Inc.

This work is licensed under

the Creative

Commons Attribution International

License (CC BY

4.0).

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

Open Access

Y. Y. Wang, Y. F. Zhao

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

together with a sequence of norms

( )

⋅

the

nn×

matrix space

( )

for each

∈ 

, which satisfies the following

Ruan’s axioms OI, OII:

( ) ( )

{ }

OI : max , ;

mn m n







 

( ) ( )

OII :

αβ α β

≤

for all

( ) ( )

vMVwMV∈∈

and

( ) ( )

nm mn

αβ

∈∈

. If

is an (ab-

stract) operator space, then there is a complete isometry

from

( )



that is,

( )

,,ij n ij



Ψ=





for all

( )

ij n

v MVn



∈∈





An abstract numerical radius operator space is a complex linear space

to-

gether with a sequence of norms

( )

⋅

on the

nn×

matrix space

( )

for each

n∈ 

, which satisfies the following axioms WI, WII:

( ) (

)

{ }

WI : max , ;

mn m n







 

( )

WII :

αα α

≤



for all

( ) ( )

vMVwMW∈∈

and

( )

∈ 

. Let

( )

⋅

be the numerical

radius norm on

( )



. If

is an abstract numerical radius operator space, then

there is a



-complete isometry

from

( )

V 

( )



, that is,

( )

,,n ij n ij



Φ=





for all

( )

ij n

v MVn



∈∈





. Given a numerical ra-

dius operator

( )

V 

, we can define an operator space

( )



( )

OW v









for all

( )

vMV∈

Y. Y. Wang, Y. F. Zhao

DOI:

10.4236/jamp.2019.76085 1253

Journal of Applied Mathematics and Physics

Given abstract numerical radius operator spaces (or operator spaces)

,VW

and a linear map

from

( )

is defined

to be

( )

,n ij





for each

( )

ij n

v MVn



∈∈





. We use a simple notation for

the norm of

( )

,ij n

v v MV



= ∈



to be

( )

v

(resp.

( )

v

) instead of

( )

v

(resp.

( )

v

), and for the norm of

( )

f MV∈

to be

(

) (

) ( ) ( )

{ }

sup : , 1

ij n

f fv v v M V v



==∈≤





We denote the norm

( )

( ) ( )

{ }

sup : , 1

n n ij n

v v v MV v

ϕϕ



= =∈≤



 

(resp.

(

)

( )

{ }

sup : , 1

n n ij n

v x v MV v

ϕϕ



= =∈≤



 

The



-completely bounded norm (resp. completely bounded norm) of

is de-

fined to be

( )

{ }

sup :

ϕϕ

= ∈ 

, (resp.

( )

{ }

sup :

ϕϕ

= ∈ 

We say



-completely bounded (resp. completely bounded) if

( )

<∞

(resp.

( )

<∞

), and



-completely contractive (resp.

completely contractive) if

( )

≤

(resp.

(

)

≤

). We call

is a



-complete isometry (resp. complete isometry) if

( )

=

(resp.

( )

=

) for each

( )

xMVn∈∈

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

, 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

( )

,mn

matrix space over

( )

C Ω

Space of continuous complex functions vanishing at

∞

Ω

Matrix norm of

,vw

Scalar pairing of matrices

,vw

Matrix pairing of matrices

( )

V 

Operator space

( )

V 

Numerical radius operator space

( )

⋅

Numerical radius norm on

(

)



( )

,BVW

Space of bounded mappings

( )

,B VW

Space of w*-bounded mappings

( )

,CB V W

Space of completely bounded mappings

( )

,CB V W

Space of



-completely bounded mappings

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