A Note on Numerical Radius Operator Spaces

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.


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 injective. 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 operator 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 We denote the norm n ϕ by The  -completely bounded norm (resp. completely bounded norm) of ϕ is defined to be , and ϕ is  -completely contractive (resp. completely contractive) if In Section 2, we study the bounded maps on finite dimension numerical radius operators and commutation C*-algebras. We prove these maps are all  -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  . In Section 4, we define the numerical radius operator spaces In order to improve the readability of the paper, we give an index of notation:

Bound Linear Maps
In this section, we study some bounded linear maps on the numerical radius operator spaces. : Proof. Since , by Lemma 3.8 and 3.9 in [18], we have Now we consider the condition for finite dimensional numerical radius operator spaces.
Proof. Let us suppose that W has dimension n. We may select an Auerbach basis for W, which by definition is a vector basis 1 2 , , , n w w w  with : : : : : : : the result follows.  For any commutative C*-algebra, we can assume that  coincides with C 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- Proof. We can assume that  coincides with C Ω . Taking the supremum over all w ∈ Ω and n α ∈  with and thus letting α also stand for column matrices, This shows that that for all n ∈  , and thus

Dual Spaces of Numerical Radius Operator Spaces
In this section, we introduce a lemma first.
Proof. If we are given , then we may use the Hahn-Banach theorem to find a linear functional [18], there is a corresponding  -complete contraction : The reverse inequality is trivial.  There is a natural numerical radius operator space structure on the mapping space . In this paper, we consider the dual space Our task is to define determines a linear mapping :    : is a  -complete isometry. It is obvious that the mapping Φ is continuous in the weak* topology. Since is also weak* compact and is a closed subspace of ( ) ( ) ,ω   . Finally, Φ is Journal of Applied Mathematics and Physics one-to-one and weak* continuous on , for any 0 ε > there exist a sufficiently large integer

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 morphisms being the  -completely bounded mappings. We have a natural "forgetful" functor : N η → D which maps a numerical radius into its underlying normed space. We say that a functor : , and for each bounded linear mapping of normed space : E F ϕ → , the corresponding mapping ( ) ( ) ( ) For any Banach space E, we let . We define the matrix norms : 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 .   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

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 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.

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

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.