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  discovery of an analogue of the Hahn-Banach theorem. Since the discovery of an abstract characterization of operator space by Ruan , there have been many more applications of operator space to other branches in functional analysis. Effros and Ruan studied the mapping spaces in  and the minimal and maximal operator spaces in . The fundamental and systematic developments in the theory of tensor product of operator spaces can be found in  . The tensor products provide a fruitful approach to mapping spaces and local property. For example, Effros, Ozawa and Ruan  showed that an operator space V is nuclear if and only if V is locally reflexive and is injective. Dong and Ruan  showed that an operator space V is exact if and only if V is locally reflexive and is weak* exact. In , Han showed that an operator space V satisfies condition C if and only if it satisfies conditions and . Based on the work of Han, Wang  gave a characterization of condition on the operator spaces. Amini, Medghalchi and Nikpey  proved that an injective operator space is global exactness if and only if it is reflexive. The readers may refer to   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  . In , 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  . 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 operator 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 numerical 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 on the matrix space for each , which satisfies the following Ruan’s axioms OI, OII:
for all and . If V is an (abstract) operator space, then there is a complete isometry from V to , that is, for all .
An abstract numerical radius operator space is a complex linear space V together with a sequence of norms on the matrix space for each , which satisfies the following axioms WI, WII:
for all and . Let be the numerical radius norm on . If V is an abstract numerical radius operator space, then there is a -complete isometry from to , that is, for all . Given a numerical radius operator , we can define an operator space by
for all .
Given abstract numerical radius operator spaces (or operator spaces) and a linear map from V to W, from to is defined to be for each . We use a simple notation for the norm of to be (resp. ) instead of (resp. ), and for the norm of to be
We denote the norm by
The -completely bounded norm (resp. completely bounded norm) of is defined to be , (resp. ). We say is -completely bounded (resp. completely bounded) if (resp. ), and is -completely contractive (resp. completely contractive) if (resp. ). We call is a -complete isometry (resp. complete isometry) if (resp. ) for each .
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 and for a normed space E, and prove that and .
In order to improve the readability of the paper, we give an index of notation:
2. Bound Linear Maps
In this section, we study some bounded linear maps on the numerical radius operator spaces.
Proposition 2.1. If is an operator space and is a numerical radius operator space satisfies , then the mapping
is -completely isometric.
Proof. Since , by Lemma 3.8 and 3.9 in , we have
Now we consider the condition for finite dimensional numerical radius operator spaces.
Proposition 2.2. Given abstract operator spaces and with either V or W n-dimensional, and are numerical radius operator spaces, any linear mapping satisfies
Proof. Let us suppose that W has dimension n. We may select an Auerbach basis for W, which by definition is a vector basis with , there exist with and . Since
where are -complete isometries from to W, and are bounded linear functionals on V. It follows from Lemma 2.3 in  that
Similarly, if V is n-dimensional, then we may replace W by , which has dimension less than or equal to n, and the result follows from the previous argument.
Proposition 2.3. If and are n-dimensional operator spaces, , are numerical radius operator spaces, then there exists a linear isomorphism such that
Proof. We choose Auervach bases and , together with dual bases and with . We have that
are inverse linear mappings. Since
the result follows.
For any commutative C*-algebra, we can assume that coincides with. We may identify with. When given, we define
then 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 linear mapping satisfies
Proof. We can assume that coincides with. Taking the supremum over all and with, we have
and thus letting also stand for column matrices,
This shows that that for all, and thus .
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, there exists a -complete contraction such that.
Proof. If we are given, then we may use the Hahn-Banach theorem to find a linear functional with and. From Lemma 2.4 in , there is a corresponding -complete contraction for which
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 by introducing an appropriate norm on.
Each determines a linear mapping, where. This gives us a linear isomorphism, which we use to determine the norm on. Thus, if we let be the corresponding normed space, we have the isometric identification
For any, we have from Lemma 2.3 in ,
where is the matrix pairing. Conversely, the norm on determines that on. Since we have from Lemma 3.1 that for any,
Proposition 3.2. The matrix norms on determine a numerical radius operator space.
Proof. Let us suppose that we are given,. Then
and hence. We have WII.
On the other hand, given, and with,
and hence. We have WI.
If is -completely bounded mapping, then we let be the dual Banach space mapping. For any and, we have
Proposition 3.3. Given numerical radius operator spaces V and W, and a -completely bounded mapping, we have for all, and.
Proof. The second relation is immediate from the first. The first follows from the calculation
where the supermum is taken over all and of norm less than 1.
We also note that given a -completely bounded mapping, its second adjoint mapping is in with, where restricted to V is equal to.
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 mapping is a dual realization of W on, if it is a weak* homeomorphic -completely isometric injection.
Theorem 3.4. If V is a complete numerical radius operator space, then has a dual realization on a Hilbert space.
Proof. Let. We have from Lemma 2.3  that if, then. We define and we let, where is the integer with. The argument in the proof of Theorem 2.1 in  shows that the mapping
is a -complete isometry. It is obvious that the mapping is continuous in the weak* topology. Since is weak* compact, then its domain is also weak* compact and is a closed subspace of. Finally, is one-to-one and weak* continuous on, thus it is a weak* homeomorphism. Since V is complete, maps weak* homeomorphically onto its image.
Proposition 3.5. If W is complete, then so is.
Proof. Let us suppose that W is complete. It suffices to show that is a closed subspace of. Given any Cauchy sequence, it is clear that is a Cauchy sequence in. From classical Banach space theory, is complete, and thus there is a bounded linear mapping such that converges to in the norm topology, i.e.,. Since is Cauchy in, for any there exist a sufficiently large integer such that whenever, we have
Given any and, we have
Since converges to in W, we have
and thus. It follows that and converges to in.
4. The Min and Max Numerical Radius Operator Spaces
We let 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 be the category of numerical radius operator spaces with the morphisms being the -completely bounded mappings. We have a natural “forgetful” functor which maps a numerical radius into its underlying normed space. We say that a functor is a strict quantization if for each normed space E, , and for each bounded linear mapping of normed space, the corresponding mapping satisfies.
For any Banach space E, we let and . We define the matrix norms and for by
Proposition 4.1. and are both numerical radius operator spaces.
Proof. To see that these are indeed numerical radius operator space matrix norms, it suffices to consider the linear injections
respectively. We have the natural numerical radius operator space identifications and.
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, then it follows from Lemma 2.3 in  that
Since, we conclude that for any.
Proposition 4.2. For any numerical radius operator space V and normed space E, and any linear mapping, we have
Proof. Let us suppose that and. Then
But implies that
and thus for all. The inversion is clear.
If is a contraction, then since is a contraction,
is -completely contractive. We conclude that is a strict quantization functor. If is an isometric injection, then it follows that is -completely isometric since we may extend any to a functional.
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, then. For any and, we have
From the above, we conclude that.
In particular, if we are given normed spaces E and F and a contraction, since is a contraction, we have
is a -complete contraction. Thus is a strict quantization.
If there is a contraction such that, then is -completely isometric since. This is also the case for the canonical injection, since any contraction automatically extends to the contraction.
Proposition 4.4 If D is a subset of, and the absolutely convex hull is weak* dense in. Then for any,
Proof. Let us suppose that for all. If where and, then
For the absolutely convex hull is weak* dense in, given an arbitrary element, we may find a net converging to g in the weak* topology. Then converges to in the numerical radius norm topology. It follows that, and thus.
For any, the linear mappings are just the weak* linear mappings from into, and thus we have the isometric identification.
Theorem 4.5. Suppose E is a normed space, then.
Proof. Given a normed space E, and a linear mapping, the second adjoint provides an extension of f to a weak* continuous mapping from to. This provides us with a natural identification . Thus, we have the isometries
The result follows.
If is a locally compact Hausdorff space and is the corresponding commutative C*-algebra, then we have a natural mapping
It is a simple consequence of the bipolar theorem that is weak* dense in. From our preceding observation, if is an element of, we have
i.e.,. We conclude that as a numerical radius operator space, Z is just the minimal quantization of its underlying Banach space, i.e.,.
Theorem 4.6. Suppose E is a normed space, then.
Proof. Given a normed space E, and an isometric injection, where Z is a commutative C*-algebra. We have a corresponding commutative diagram
where the first column is an isometry, the second column is a -complete isometry, and both rows are isometric. Since is a numerical radius operator space, it determines the minimal numerical radius operator space structure on, hence. Thus, we have the -complete isometries
and since these identifications are compatible with the dualities, we have the -complete isometry.
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).
 Pisier, G. (2003) Introduction to Operator Space Theory. London Mathematical Society Lecture Note Series, Vol. 294. Cambridge University Press, Cambridge.
 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.
 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.