We establish a relationship between Schreiner's matrix regular operator space
and Werner's (nonunital) operator system. For a matrix ordered operator space
$V$ with complete norm, we show that $V$ is completely isomorphic and complete
order isomorphic to a matrix regular operator space if and only if both $V$ and
its dual space $V^*$ are (nonunital) operator systems.Comment: 8 pages, to appear in JMA

Han, Kyung Hoon

English

arXiv

AbstractWe establish a relationship between Schreiner's matrix regular operator space and Werner's (nonunital) operator system. For a matrix ordered operator space V with complete norm, we show that V is completely isomorphic and complete order isomorphic to a matrix regular operator space if and only if both V and its dual space V∗ are (nonunital) operator systems

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Matrix regular operator space and operator system 

J. Math. Anal. Appl. 367 (2010) 516–521Contents lists available at ScienceDirect
Journal of Mathematical Analysis and
Applications
www.elsevier.com/locate/jmaa
Matrix regular operator space and operator system
Kyung Hoon Han
Department of Mathematical Sciences, Seoul National University, San 56-1 ShinRimDong KwanAk-Gu, 151-747 Seoul, Republic of Korea
a r t i c l e i n f o a b s t r a c t
Article history:
Received 28 October 2009
Available online 6 February 2010
Submitted by D. Blecher
Keywords:
Matrix regular operator space
Operator system
We establish a relationship between Schreiner’s matrix regular operator space and
Werner’s (nonunital) operator system. For a matrix ordered operator space V with
complete norm, we show that V is completely isomorphic and complete order isomorphic
to a matrix regular operator space if and only if both V and its dual space V ∗ are
(nonunital) operator systems.
© 2010 Elsevier Inc. All rights reserved.
1. Introduction
The theory of operator spaces has been developed as a noncommutative counterpart of the theory of Banach spaces.
The order structures of classical Banach spaces have been studied mostly under the Banach lattice framework. However,
the most basic examples of operator algebras such as Mn (n  2) do not possess these lattice structures (consider 2 × 2
matrices 02 and E1,2 + E2,1 and the family λE1,1 + λ−1E2,2 for λ > 0). Hence, it is natural to consider order structures that
will work in a noncommutative setting. In this paper, we focus on such order structures of operator spaces.
Two of the several basics of operator space theory are Ruan’s representation theorem and the duality: every operator
space can be embedded into B(H) completely isometrically and there is a natural operator space structure on the Banach
dual of an operator space. From this standpoint, there are two deﬁnitions of the order structures of operator spaces. Werner’s
(nonunital) operator system corresponds to the representation and Schreiner’s matrix regular operator space to the duality.
Usually, an operator system means a unital involutive subspace of B(H) or its abstract characterization given by Choi and
Effros [3], but here we follow Werner’s terminology. In this paper, a (nonunital) operator system means a matrix ordered
operator space which is completely isomorphic and complete order isomorphic to an involutive subspace of B(H) or its
abstract characterization given by Werner [12]. For a matrix ordered operator space V with complete norm, V is matrix
regular if and only if its dual space V ∗ is matrix regular [11].
The category of (nonunital) operator systems contains the class of C∗-algebras and Haagerup’s noncommutative
Lp-spaces [6]. The category of matrix regular operator spaces contains the class of C∗-algebras and their duals, preduals
of von Neumann algebras, and the Schatten class Sp [10].
Karn proved that every matrix regular operator space is a (nonunital) operator system [7]. Since the dual space of a
matrix regular operator space is matrix regular, the dual space of a matrix regular operator space is also a (nonunital)
operator system. Its converse would be reasonable in the completely isomorphic context because Werner’s (nonunital)
operator system is deﬁned not in a completely isometric sense but in a completely isomorphic sense. The purpose of
this paper is to show that a matrix ordered operator space V with complete norm is completely isomorphic and complete
order isomorphic to a matrix regular operator space if and only if both V and its dual space V ∗ are (nonunital) operator
systems.
E-mail address: kyunghoon.han@gmail.com.0022-247X/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2010.01.061
K.H. Han / J. Math. Anal. Appl. 367 (2010) 516–521 5172. Preliminaries
Recall that a complex vector space V is matrix ordered if
(1) V is a ∗-vector space (hence so is Mn(V ) for all n 1),
(2) each Mn(V ), n 1, is partially ordered by a (not necessarily proper) cone Mn(V )+ ⊂ Mn(V )sa , and
(3) if α ∈ Mm,n , then α∗Mm(V )+α ⊂ Mn(V )+ .
Here the positive cone need not be proper, in other words, it may be the case that Mn(V )+ ∩ −Mn(V )+ = {0}. The reason
to exclude the proper condition is due to the fact that the dual cone of a proper cone need not be proper.
An operator space V is called a matrix ordered operator space iff V is a matrix ordered vector space and for every n ∈ N,
(1) the ∗-operation is an isometry on Mn(V ), and
(2) the cones Mn(V )+ are closed.
For a matrix ordered operator space V and its dual space V ∗ , the positive cone on Mn(V ∗) for each n ∈ N is deﬁned by
Mn
(
V ∗
)+ = CB(V ,Mn) ∩ CP(V ,Mn).
Then the operator space dual V ∗ with this positive cone is a matrix ordered operator space [11, Corollary 3.2].
For a matrix ordered operator space V with complete norm, we say that V is a matrix regular operator space if for each
n ∈ N and for all v ∈ Mn(V )sa
(1) u ∈ Mn(V )+ and −u  v  u imply that ‖v‖n  ‖u‖n , and
(2) ‖v‖n < 1 implies that there exists u ∈ Mn(V )+ such that ‖u‖n < 1 and −u  v  u.
Due to condition (1), it is easily seen that the positive cone of a matrix regular operator space is always proper. A matrix
regular operator space can be described in another way. A matrix ordered operator space V with complete norm is matrix
regular if and only if the following condition holds: for all x ∈ Mn(V ), ‖x‖n < 1 if and only if there exist a,d ∈ Mn(V )+ ,
‖a‖n < 1 and ‖d‖n < 1, such that
( a x
x∗ d
) ∈ M2n(V )+ [11, Theorem 3.4].
The class of matrix regular operator spaces has a nice duality property. Let V be a matrix ordered operator space with
complete norm. Schreiner showed that V is matrix regular if and only if its dual space V ∗ is matrix regular [11, Corollary 4.7,
Theorem 4.10].
Let X be a matrix ordered operator space with a proper positive cone. For x ∈ Mn(X), the modiﬁed numerical radius is
deﬁned by
νX (x) = sup
{∣∣∣∣ϕ
((
0 x
x∗ 0
))∣∣∣∣ : ϕ ∈ M2n(X)∗1,+
}
.
We call a matrix ordered operator space with a proper positive cone an operator system iff there is a k > 0 such that for all
n ∈ N and x ∈ Mn(X),
‖x‖n  kνX (x).
Since we always have the inequality νX (x) ‖x‖n , we can say that an operator system is a matrix ordered operator space
such that the operator space norm and the modiﬁed numerical radius are equivalent uniformly for all n ∈ N.
Werner showed that X is an operator system if and only if there is a complete order isomorphism Φ from X onto an
involutive subspace of B(H), which is a complete topological onto-isomorphism [12, Theorem 4.15]. Hence, the operator
system is an abstract characterization of the involutive subspace of B(H) in a completely isomorphic and complete order
isomorphic sense.
3. Matrix regular operator space and operator system
Karn showed that every matrix regular operator space can be embedded into B(H) 2-completely isomorphically and
complete order isomorphically [7]. Here we give another proof of the result that is more ﬁtting to Werner’s axiomatic
framework. This idea also appears in the recent preprint [8, Theorem 2.4].
Theorem 1. Every matrix regular operator space is an operator system with a dominating constant 2.
Proof. Since the dual space of a matrix regular operator space is matrix regular and the canonical inclusion map from a
matrix ordered operator space into its bidual is a completely isometric complete order isomorphism [11, Theorem 4.9], it is
suﬃcient to show that the dual space of a matrix regular operator space is an operator system.
518 K.H. Han / J. Math. Anal. Appl. 367 (2010) 516–521Suppose that V is a matrix regular operator space. We choose an element F = [ f i j] in Mn(V ∗). Its norm can be written
as
‖F‖Mn(V ∗) = ‖F‖CB(V ,Mn)
= sup{∥∥Fn(x)∥∥Mn2 : x ∈ Mn(V )‖·‖<1}
= sup
{∥∥∥∥
(
0 Fn(x)
Fn(x)∗ 0
)∥∥∥∥
M2n2
: x ∈ Mn(V )‖·‖<1
}
= sup
{∣∣∣∣
〈(
0 Fn(x)
Fn(x)∗ 0
)
ξ |ξ
〉∣∣∣∣ : x ∈ Mn(V )‖·‖<1, ξ ∈ (22n2)1
}
.
We choose elements x in Mn(V )‖·‖<1 and ξ in (22n2 )1. By [11, Theorem 3.4], there exist a, d in Mn(V ) such that ‖a‖n < 1,
‖d‖n < 1 and
( a x
x∗ d
) ∈ M2n(V )+ . Then we have ∥∥( a xx∗ d)∥∥2n < 2. We deﬁne a linear functional ϕx,ξ : M2n(V ∗) → C by
ϕx,ξ (G) = 1
2
〈(
In2 0 0 0
0 0 0 In2
)
G2n
((
a x
x∗ d
))⎛⎜⎝
In2 0
0 0
0 0
0 In2
⎞
⎟⎠ ξ |ξ
〉
, G ∈ M2n
(
V ∗
)= CB(V ,M2n).
Then ϕx,ξ is a positive contractive functional. Putting G =
( 0 F
F ∗ 0
)
, we get
ϕx,ξ
((
0 F
F ∗ 0
))
= 1
2
〈(
In2 0 0 0
0 0 0 In2
)⎛⎜⎝
0 0 Fn(a) Fn(x)
0 0 Fn(x∗) Fn(d)
F ∗n (a) F ∗n (x) 0 0
Fn(x)∗ F ∗n (z) 0 0
⎞
⎟⎠
⎛
⎜⎝
In2 0
0 0
0 0
0 In2
⎞
⎟⎠ ξ |ξ
〉
= 1
2
〈(
0 Fn(x)
Fn(x)∗ 0
)
ξ |ξ
〉
.
It follows that
‖F‖Mn(V ∗) = 2 sup
{∣∣∣∣ϕx,ξ
((
0 F
F ∗ 0
))∣∣∣∣ : x ∈ Mn(V )‖·‖<1, ξ ∈ (22n)1
}
 2νV ∗(F ),
where νV ∗(F ) denotes the modiﬁed numerical radius of F . 
In general, the embedding cannot be chosen completely isometrically and complete order isomorphically as can be seen
from the two-dimensional L1-space 12 [2, Proposition 1.1]. In the case of the Schatten class Sp , the constant can be chosen
to be 2
1
p [6].
The direct converse of Theorem 1 is false. If we consider the operator system{(
0 α
β 0
)
∈ M2n: α,β ∈ Mn
}
,
then its positive cone is trivial, thus the second condition on matrix regularity cannot be satisﬁed for any operator system
complete order isomorphic to the above one. Because the dual space of a matrix regular operator space is also matrix
regular, the matrix regularity of V implies that both V and its dual space V ∗ are operator systems. Our next goal is to
prove the converse in a completely isomorphic and complete order isomorphic sense. Informally, the ﬁrst and the second
conditions on matrix regularity imply that the positive cone is small and large, respectively, in some sense. In other words,
we can say that the positive cone of a matrix regular operator space is just the right size. The deﬁnition of an operator
system means the dual cone is large enough, or equivalently, that the positive cone of an operator system is small enough.
As we have just seen, the positive cone of an operator system may be trivial. We see from these informal observations that
it is natural to consider the problem set forth as the goal of the present paper.
Lemma 2. Suppose that V is a matrix ordered operator space with complete norm satisfying the following two conditions:
(1) for all x, y ∈ Mn(V )sa,−y  x y implies ‖x‖n  ‖y‖n,
(2) for all x ∈ Mn(V )sa with ‖x‖ < 1, there exist a,d ∈ Mn(V )+ such that ‖a‖n,‖d‖n < K and
( a x
x∗ d
) ∈ M2n(V )+ .
Then V is K -completely isomorphic and complete order isomorphic to a matrix regular operator space.
K.H. Han / J. Math. Anal. Appl. 367 (2010) 516–521 519Proof. We ﬁrst deﬁne
‖x‖reg = inf
{
max
{‖a‖n,‖d‖n}:
(
a x
x∗ d
)
∈ M2n(V )+
}
, x ∈ Mn(V ).
Note that this deﬁnition is similar to the norm ‖ · ‖dec of a decomposable map [5]. The set which we take an inﬁmum over
is not empty and we have ‖x‖reg  K‖x‖n . Multiplying both sides by the scalar matrix
( 1 0
0 −1
)
, we see that(
a x
x∗ d
)
∈ M+2n if and only if −
(
a 0
0 d
)

(
0 x
x∗ 0
)

(
a 0
0 d
)
.
The inequality ‖x‖n  ‖x‖reg follows from condition (1). We choose elements
( a1 x
x∗ d1
)
and
( a2 y
y∗ d2
)
in M2n(V )+ . Since( a1+a2 x+y
(x+y)∗ d1+d2
)
belongs to M2n(V )+ , we have
‖x+ y‖reg max
{‖a1 + a2‖n,‖d1 + d2‖n}
=
∥∥∥∥
(
a1 + a2 0
0 d1 + d2
)∥∥∥∥
2n

∥∥∥∥
(
a1 0
0 d1
)∥∥∥∥
2n
+
∥∥∥∥
(
a2 0
0 d2
)∥∥∥∥
2n
= max{‖a1‖n,‖d1‖n}+max{‖a2‖n,‖d2‖n}.
It follows that ‖x+ y‖reg  ‖x‖reg + ‖y‖reg . For λ = eit |λ| ∈ C, we have( |λ|a λx
(λx)∗ |λ|b
)
=
(
eit |λ| 12 0
0 |λ| 12
)(
a x
x∗ d
)(
e−it |λ| 12 0
0 |λ| 12
)
∈ M2n(V )+.
It follows that ‖λx‖reg = |λ|‖x‖reg . Because we have ‖x‖n  ‖x‖reg for all x ∈ Mn(V ), ‖x‖reg = 0 implies x = 0. Hence, ‖ · ‖reg
is a norm on Mn(V ) for each n ∈ N.
Next let us show that ‖αxβ‖reg  ‖α‖‖x‖reg‖β‖ for x ∈ Mm(V ), α ∈ Mn,m , β ∈ Mm,n . To this end, we may assume that
‖α‖ = ‖β‖. For ( a xx∗ d) ∈ M2m(V )+ , we have(
αaα∗ αxβ
(αxβ)∗ β∗dβ
)
=
(
α 0
0 β∗
)(
a x
x∗ d
)(
α∗ 0
0 β
)
∈ M2n(V )+.
It follows that
‖αxβ‖reg max
{∥∥αaα∗∥∥n,∥∥β∗dβ∥∥n}
 ‖α‖‖β‖max{‖a‖m,‖d‖m},
thus ‖αxβ‖reg  ‖α‖‖β‖‖x‖reg . Suppose that(
a1 x
x∗ d1
)
∈ M2m(V )+ and
(
a2 y
y∗ d2
)
∈ M2n(V )+
with ‖a1‖m,‖d1‖m < ‖x‖reg + ε and ‖a2‖n,‖d2‖n < ‖y‖reg + ε. Then we have⎛
⎜⎝
a1 0 x 0
0 a2 0 y
x∗ 0 d1 0
0 y∗ 0 d2
⎞
⎟⎠=
⎛
⎜⎝
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
⎞
⎟⎠
⎛
⎜⎝
a1 x 0 0
x∗ d1 0 0
0 0 a2 y
0 0 y∗ d2
⎞
⎟⎠
⎛
⎜⎝
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
⎞
⎟⎠ ∈ M2(m+n)(V )+.
It follows that
‖x⊕ y‖reg max
{‖a1‖m,‖a2‖n,‖d1‖m,‖d2‖n}< max{‖x‖reg,‖y‖reg}+ ε.
Hence, (V ,‖ · ‖reg) is an operator space.
For
( a x
x∗ d
) ∈ M2n(V )+ , we have(
d x∗
x a
)
=
(
0 1
1 0
)(
a x
x∗ d
)(
0 1
1 0
)
∈ M2n(V )+,
thus ‖x‖reg = ‖x∗‖reg . Since the identity map id : (V ,‖ · ‖) → (V ,‖ · ‖reg) is a K -complete isomorphism, the operator space
(V ,‖ · ‖reg) is complete and the positive cone Mn(V )+ is closed with respect to the norm ‖ · ‖reg . For a ∈ Mn(V )+ , we have
520 K.H. Han / J. Math. Anal. Appl. 367 (2010) 516–521(
a a
a a
)
=
(
1
1
)
a (1 1 ) ∈ M2n(V )+,
thus ‖a‖reg = ‖a‖n . If an element x belongs to Mn(V ) with ‖x‖reg < 1, then there exist elements a and d in Mn(V )+
such that ‖a‖n , ‖d‖n < 1 and
( a x
x∗ d
) ∈ M2n(V )+ . Since a and d are positive, we have ‖a‖reg , ‖d‖reg < 1. The con-
verse is obvious. By [11, Theorem 3.4], (V ,‖ · ‖reg, {Mn(V )+}n∈N) is a matrix regular operator space. The identity map
id : (V ,‖ · ‖, {Mn(V )+}n∈N) → (V ,‖ · ‖reg, {Mn(V )+}n∈N) is a K -completely isomorphic complete order isomorphism. 
Lemma 3. Suppose that a matrix ordered operator space V and its dual space V ∗ are operator systems. Then the dual space V ∗ is
completely isomorphic and complete order isomorphic to a matrix regular operator space.
Proof. By looking at the dual space V ∗ as a ∗-subspace of B(K ), we can say that condition (1) of Lemma 2 is satisﬁed.
Suppose that W is a ∗-subspace of B(H) and Φ : W → V is a completely isomorphic complete order isomorphism with
‖Φ‖cb  1. We put
X =
{(
λIH x
y μIH
)
: λ,μ ∈ C, x, y ∈ W
}
.
Then X is a unital operator system in B(H2). We take an element F in Mn(V ∗) with ‖F‖Mn(V ∗) < 1. By [9, Lemma 8.1], the
linear map ϕ : X → M2n deﬁned by
ϕ
((
λIH x
y μIH
))
=
(
λIn F ◦ Φ(x)
F ∗ ◦ Φ(y) μIn
)
is a unital completely positive map. By Arveson’s extension theorem [1], ϕ has a unital completely positive extension
ψ : M2(W ) + CIH ⊕ CIH → M2n. The linear map
θ : x ∈ W →
(
x x
x x
)
=
(
1
1
)
x (1 1 ) ∈ M2(W )
is completely positive. We write
ψ ◦ θ =
(
ϕ1 F ◦ Φ
F ∗ ◦ Φ ϕ2
)
∈ M2n
(
W ∗
)+
, ‖ϕ1‖Mn(W ∗)  1, ‖ϕ2‖Mn(W ∗)  1.
Then we have(
ϕ1 ◦ Φ−1 F
F ∗ ϕ2 ◦ Φ−1
)
∈ M2n
(
V ∗
)+
and
∥∥ϕ1 ◦ Φ−1∥∥cb,∥∥ϕ2 ◦ Φ−1∥∥cb  ∥∥Φ−1∥∥cb.
By Lemma 2, we conclude that the dual space V ∗ is completely isomorphic and complete order isomorphic to a matrix
regular operator space. 
Theorem 4. Suppose that both V and its dual space V ∗ are operator systems with complete norm. Then V is completely isomorphic
and complete order isomorphic to a matrix regular operator space.
Proof. Once again we consider the space V as a ∗-subspace of B(H) and see that condition (1) of Lemma 2 is satisﬁed.
By Lemma 3, there exist a matrix regular operator space W and a completely isomorphic complete order isomorphism
Φ : V ∗ → W . For x ∈ Mn(V ), we deﬁne
‖x‖Mn(W∗) = sup
{∣∣ϕ(x)∣∣: ϕ ∈ Mn(V )∗, ∥∥Φ(ϕ)∥∥Tn(W )  1}.
Endowing V with this matrix norm ‖ · ‖W∗ , we get an operator space predual of W . Since the two norms ‖ · ‖Mn(W∗) and‖ · ‖Mn(V ) are equivalent, the positive cone Mn(V )+ is closed with respect to the norm ‖ · ‖Mn(W∗) and the operator space
(V ,‖ · ‖W∗ ) is complete. Applying the conjugate linear variation of [4, Theorem 4.1.8] to the involution, we get∥∥x∗∥∥Mn(W∗) = sup{∣∣ϕ(x∗)∣∣: ∥∥Φ(ϕ)∥∥Tn(W )  1}
= sup{∣∣ϕ(x)∣∣: ∥∥Φ(ϕ)∗∥∥Tn(W )  1}
= ‖x‖Mn(W∗).
Hence the space W∗ := (V , {‖ · ‖Mn(W∗)}n∈N, {Mn(V )+}n∈N) is a matrix ordered operator space with a complete norm, and
its dual space is W . Since the predual of a matrix regular operator space is also matrix regular, W∗ is matrix regular. The
predual map Φ∗ : W∗ → V is completely isomorphic and complete order isomorphic. 
Corollary 5. For a matrix ordered operator space V with complete norm, V is completely isomorphic and complete order isomorphic
to a matrix regular operator space if and only if both V and its dual space V ∗ are operator systems.
K.H. Han / J. Math. Anal. Appl. 367 (2010) 516–521 521References
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