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

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 B ( H ) . The study of operator space begins with Arverson’s [

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 [

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 O n ( ⋅ ) on the n × n matrix space M n ( V ) for each n ∈ ℕ , which satisfies the following Ruan’s axioms OI, OII:

OI : O m + n ( ( v 0 0 w ) ) = max { O m ( v ) , O n ( w ) } ;

OII : O n ( α v β ) ≤ ‖ α ‖ O m ( v ) ‖ β ‖

for all v ∈ M m ( V ) , w ∈ M n ( V ) and α ∈ M n , m ( ℂ ) , β ∈ M m , n ( ℂ ) . If V is an (abstract) operator space, then there is a complete isometry Ψ from V to B ( H ) , that is, ‖ [ Ψ ( v i , j ) ] ‖ n = O n ( [ v i , j ] ) for all [ v i , j ] ∈ M n ( V ) , n ∈ ℕ .

An abstract numerical radius operator space is a complex linear space V together with a sequence of norms W n ( ⋅ ) on the n × n matrix space M n ( V ) for each n ∈ ℕ , which satisfies the following axioms WI, WII:

WI : W m + n ( ( v 0 0 w ) ) = max { W m ( v ) , W n ( w ) } ;

WII : W n ( α v α ) ≤ ‖ α ‖ 2 W m (v)

for all v ∈ M m ( V ) , w ∈ M n ( W ) and α ∈ M n , m ( ℂ ) . Let ω ( ⋅ ) be the numerical radius norm on B ( H ) . If V is an abstract numerical radius operator space, then there is a W -complete isometry Φ from ( V , W n ) to ( B ( H ) , ω n ) , that is, ω n ( Φ ( v i , j ) ) = W n ( [ v i , j ] ) for all [ v i , j ] ∈ M n ( V ) , n ∈ ℕ . Given a numerical radius operator ( V , W n ) , we can define an operator space ( V , O n ) by

O W : 1 2 O n ( v ) = W 2 n ( ( 0 v 0 0 ) )

for all v ∈ M n ( V ) .

Given abstract numerical radius operator spaces (or operator spaces) V , W and a linear map φ from V to W, φ n from M n ( V ) to M n ( W ) is defined to be φ n ( [ v i , j ] ) for each [ v i , j ] ∈ M n ( V ) , n ∈ ℕ . We use a simple notation for the norm of v = [ v i , j ] ∈ M n ( V ) to be W ( v ) (resp. O ( v ) ) instead of W n ( v ) (resp. O n ( v ) ), and for the norm of f ∈ M n ( V ) * to be

W * ( f ) = sup { | f ( v ) | : v = [ v i , j ] ∈ M n ( V ) , W ( v ) ≤ 1 } .

We denote the norm φ n by

W ( φ n ) = sup { W ( φ n ( v ) ) : v = [ v i , j ] ∈ M n ( V ) , W ( v ) ≤ 1 }

(resp. O ( φ n ) = sup { O ( φ n ( v ) ) : x = [ v i , j ] ∈ M n ( V ) , O ( v ) ≤ 1 } ).

The W -completely bounded norm (resp. completely bounded norm) of φ is defined to be W ( φ ) c b = sup { W ( φ n ) : n ∈ ℕ } , (resp. O ( φ ) c b = sup { O ( φ n ) : n ∈ ℕ } ). We say φ is W -completely bounded (resp. completely bounded) if W ( φ ) c b < ∞ (resp. O ( φ ) c b < ∞ ), and φ is W -completely contractive (resp. completely contractive) if W ( φ ) c b ≤ 1 (resp. O ( φ ) c b ≤ 1 ). We call φ is a W -complete isometry (resp. complete isometry) if W ( φ n ( v ) ) = W ( v ) (resp. O ( φ n ( v ) ) = O ( v ) ) for each x ∈ M n ( V ) , n ∈ ℕ .

In Section 2, we study the bounded maps on finite dimension numerical radius operators and commutation C*-algebras. We prove these maps are all W -completely bounded. In Section 3, we study the dual space of a numerical radius operator space and prove its dual space has a dual realization on a Hilbert space H . In Section 4, we define the numerical radius operator spaces M i n E and M a x E for a normed space E, and prove that ( M a x E ) * = M i n E * and M a x E * = ( M i n E ) * .

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

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

Proposition 2.1. If ( V , O n ) is an operator space and ( V , W n ) is a numerical radius operator space satisfies ‖ v ‖ = 1 , then the mapping

θ v : C → V : α → α v

is W -completely isometric.

Proof. Since W max ( ℂ ) = ω ( ℂ ) , by Lemma 3.8 and 3.9 in [

W ( θ v : W max ( ℂ ) → W ( v ) ) c b ≤ O ( θ v ) c b = 1

and

O ( θ v ) c b ≤ W ( θ v : ω ( ℂ ) → W ( v ) ) c b .

So

W ( θ v : ω ( ℂ ) → W ( v ) ) c b = O ( θ v ) c b = 1. □

Now we consider the condition for finite dimensional numerical radius operator spaces.

Proposition 2.2. Given abstract operator spaces ( V , O n ) and ( W , O n ) with either V or W n-dimensional, ( V , W n ) and ( W , W n ) are numerical radius operator spaces, any linear mapping φ : V → W satisfies

W ( φ : W ( V ) → W ( W ) ) c b ≤ n W ( φ : W ( V ) → W ( W ) ) .

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 1 , w 2 , ⋯ , w n with W ( w j ) = 1 , there exist g j ∈ W ( W ) * with W ( g j ) = 1 and g j ( w i ) = δ i j . Since

i d W = ∑ j = 1 n θ w j ∘ g j .

We have

φ = ∑ j = 1 n θ w j ∘ g j ∘ φ ,

where θ w j ( α ) = α w j are W -complete isometries from ℂ to W, and g j ∘ φ are bounded linear functionals on V. It follows from Lemma 2.3 in [

W ( φ : W ( V ) → W ( W ) ) c b ≤ ∑ j = 1 n W ( θ w j : ω ( ℂ ) → W ( W ) ) c b ⋅ W ( g j ∘ φ : W ( V ) → ω ( ℂ ) ) c b = ∑ j = 1 n W ( g j ∘ φ : W ( V ) → ω ( ℂ ) ) ≤ n W ( φ : W ( V ) → W ( W ) ) .

Similarly, if V is n-dimensional, then we may replace W by φ ( W ) , which has dimension less than or equal to n, and the result follows from the previous argument. □

Proposition 2.3. If ( V , O n ) and ( W , O n ) are n-dimensional operator spaces, ( V , W n ) , ( W , W n ) are numerical radius operator spaces, then there exists a linear isomorphism φ : W ( V ) → W ( W ) such that

W ( φ : W ( V ) → W ( W ) ) c b ⋅ W ( φ − 1 : W ( W ) → W ( V ) ) c b ≤ n 2 .

Proof. We choose Auervach bases v i ∈ V and w i ∈ W ( i = 1 , ⋯ , n ) , together with dual bases f i ∈ W ( V ) * and g i ∈ W ( W ) * with W ( f i ) = W ( g i ) = 1 . We have that

φ : V → W : v ↦ ∑ i = 1 n f i ( v ) w i

and

ψ : W → V : w ↦ ∑ i = 1 n g i ( w ) v i

are inverse linear mappings. Since

and similarly

the result follows.

For any commutative C*-algebra, we can assume that

then

Theorem 2.4. Let V be a numerical radius operator space, and let

Proof. We can assume that

and thus letting

This shows that that

In this section, we introduce a lemma first.

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

Proof. If we are given

The reverse inequality is trivial.

There is a natural numerical radius operator space structure on the mapping space

Our task is to define

Each

For any

where

Proposition 3.2. The matrix norms on

Proof. Let us suppose that we are given

and hence

On the other hand, given

and hence

If

Proposition 3.3. Given numerical radius operator spaces V and W, and a

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

where the supermum is taken over all

We also note that given a

Given a numerical radius operator space W which is the dual of a complete numerical radius operator space V, and a Hilbert space

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

Proof. Let

is a

Proposition 3.5. If W is complete, then so is

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

Given any

Since

and thus

We let

For any Banach space E, we let

and

Proposition 4.1.

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

and

respectively. We have the natural numerical radius operator space identifications

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

Since

Proposition 4.2. For any numerical radius operator space V and normed space E, and any linear mapping

Proof. Let us suppose that

But

and thus

If

is

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

i.e., for any linear mapping

Proof. To prove this, it suffices to show that if

From the above, we conclude that

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

is a

If there is a contraction

Proposition 4.4 If D is a subset of

Proof. Let us suppose that

For the absolutely convex hull

For any

Theorem 4.5. Suppose E is a normed space, then

Proof. Given a normed space E,

The result follows.

If

It is a simple consequence of the bipolar theorem that

i.e.,

Theorem 4.6. Suppose E is a normed space, then

Proof. Given a normed space E, and an isometric injection

where the first column is an isometry, the second column is a

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

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 operator 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 intrinsic merit.

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

The authors declare no conflicts of interest regarding the publication of 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