EXTREME STATES, OPERATOR SPACES AND TERNARY
RINGS OF OPERATORS
A. K. Vijayarajan
Kerala School of Mathematics, Kozhikode 673 571 (India).
E-mail:vijay@ksom.res.in
ORCID:0000-0001-5084-0567
Reception: 29/09/2022 Acceptance: 14/10/2022 Publication: 29/12/2022
Suggested citation:
A.K. Vijayarajan (2022). Extrem states, operator spaces and ternary rings of operators. 3C TIC. Cuadernos de desarrollo
aplicados a las TIC,11 (2), 124-134. https://doi.org/10.17993/3ctic.2022.112.124-134
https://doi.org/10.17993/3ctic.2022.112.124-134
ABSTRACT
In this survey article on extreme states of operator spaces in
C¡
-algebras and related ternary ring of
operators an extension result for rectangular operator extreme states on operator spaces in ternary rings
of operators is discussed. We also observe that in the spacial case of operator spaces in rectangular
matrix spaces, rectangular extreme states are conjugates of inclusion or identity maps implemented by
isometries or unitaries. A characterization result for operator spaces of matrices for which the inclusion
map is an extreme state is deduced using the above mentioned results.
KEYWORDS
keyword 1, Keyword 2,...
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
124
EXTREME STATES, OPERATOR SPACES AND TERNARY
RINGS OF OPERATORS
A. K. Vijayarajan
Kerala School of Mathematics, Kozhikode 673 571 (India).
E-mail:vijay@ksom.res.in
ORCID:0000-0001-5084-0567
Reception: 29/09/2022 Acceptance: 14/10/2022 Publication: 29/12/2022
Suggested citation:
A.K. Vijayarajan (2022). Extrem states, operator spaces and ternary rings of operators. 3C TIC. Cuadernos de desarrollo
aplicados a las TIC,11 (2), 124-134. https://doi.org/10.17993/3ctic.2022.112.124-134
https://doi.org/10.17993/3ctic.2022.112.124-134
ABSTRACT
In this survey article on extreme states of operator spaces in
C¡
-algebras and related ternary ring of
operators an extension result for rectangular operator extreme states on operator spaces in ternary rings
of operators is discussed. We also observe that in the spacial case of operator spaces in rectangular
matrix spaces, rectangular extreme states are conjugates of inclusion or identity maps implemented by
isometries or unitaries. A characterization result for operator spaces of matrices for which the inclusion
map is an extreme state is deduced using the above mentioned results.
KEYWORDS
keyword 1, Keyword 2,...
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
125
1 INTRODUCTION
Arveson’s extension theorem [3, Theorem1.2.3] for completely positive(CP) maps in the context of
operator systems in
C¡
-algebras is a remarkable result in the study of boundary representations of
C¡
-algebras for operator systems. The theorem asserts that any CP map on an operator system
in
a
C¡
-algebra
to
B
.
/can be extended to a CP map from
to
B
.
/. A natural non-self-adjoint
counterpart of the set up Arveson worked with can be considered to be consisting of operator spaces
in ternary rings of operators(TROs) and completely contractive maps on them. An important recent
work in this scenario is [10] where rectangular matrix convex sets and boundary representations of
operator spaces were introduced. An operator space version of Arveson’s conjecture, namely, every
operator space is completely normed by it’s boundary representations is also established in [10].
Apart from Arveson’s fundamental work [3
–
5], we refer to the work of Douglas Farenick on extremal
theory of matrix states on operator systems [8,9] and Kleski’s work on pure completely positive maps
and boundary representations for operator systems [12].
In this article we study an extension result for rectangular extreme states on operator spaces in
TROs. A characterization result for rectangular extreme state on operator spaces of matrices with
trivial commutants is deduced.
2 Preliminaries
In this section we recall the fundamental notions that we require for the discussions later on in this
article.
Let
H
and
K
be Hilbert spaces and
B
.
H,K
/be the space of all bounded operators from
H
to
K
.
When
K
=
H
, we denote
B
.
H,H
/by
B
.
H
/. A (concrete) operator space is a closed subspace of the
(concrete)
C¡
-algebra
B
.
H
/. The space
B
.
H,K
/can be viewed as a subspace of
B
.
KH
/, and hence it
is always an operator space which we study more in later parts of this article.
An abstract characterisation of operator spaces was established by Ruan [16]. A subclass of operator
spaces called ternary rings of operators referred to as TROs is of special interest to us here. These
were shown to be the injective objects in the category of operator spaces and completely contractive
linear maps by Ruan [17].
Definition 1. A ternary ring of operators (TRO)
T
is a subspace of the
C¡
-algebra
B
.
H
/of all
bounded operators on a Hilbert space
H
that is closed under the triple product .
x, y, z
/
→xy¡z
for all
x, y, z ËT.
A triple morphism between TROs is a linear map that preserves the triple product.
A triple morphism between TROs can be seen as the top-right corner of a *-homomorphism between
the corresponding linking algebras [11]; see also [6, Corollary 8.3.5].
Clearly, an obvious, but important example of a TRO is the space
B
.
H,K
/, where
H
and
K
are
Hilbert spaces.
Definition 2. Let
X
and
Y
be operator spaces. A linear map
ϕ
:
X→Y
is called completely
contractive (CC) if the linear map nϕ:nX→nYis contractive for all n.
We denote the set of all CC maps from Xto B.H,K/by CC.X, B.H,K//.
Definition 3. A representation of a TRO
T
is a triple morphism
ϕ
:
T→B
.
H,K
/for some Hilbert
spaces Hand K.
A representation
ϕ
:
T→B
.
H,K
/is irreducible if, whenever
p
,
q
are projections in
B
.
H
/and
B
.
K
/respectively, such that
qϕ
.
x
/=
ϕ
.
x
/
p
for every
x
Ë
T
, one has
p
=0and
q
=0, or
p
=1and
q=1.
https://doi.org/10.17993/3ctic.2022.112.124-134
A linear map
ϕ
:T
→B
.
H,K
/is nondegenerate if, whenever
p
,
q
are projections in
B
.
H
/and
B.K/, respectively, such that qϕ.x/=ϕ.x/p=0for every xËT, one has p=0and q=0.
Definition 4. A rectangular operator state on an operator space
X
is a non degenerate linear map
ϕ:X→B.H,K/such that ϕcb =1where the norm here is the completely bounded norm.
Several characterizations of nondegenerate and irreducible representations of TROs are obtained
in [7, Lemma 3.1.4 and Lemma 3.1.5].
A concrete TRO T
B
.
H,K
/is said to act irreducibly if the corresponding inclusion representation
is irreducible.
3 systems and spaces
An operator system in a
C¡
-algebra is a unital selfadjoint closed linear subspace. Let
be an operator
system in a C¡-algebra , and be any other C¡-algebra.
A linear map
ϕ
:
→
is called completely positive (CP) if the linear map
nϕ
:
n→n
is
positive for all natural numbers n.
We denote the set of all unital CP maps from to B.H/by UCP.,B.H//.
One of the crucial theorems in this context is Arveson’s extension theorem [3, Theorem1.2.3].
Arveson’s extension theorem asserts that any CP map on an operator system
to
B
.
H
/can be
extended to a CP map from the C¡-algebra to B.H/.
Given an operator space
XB
.
H,K
/, we can assign an operator system
S
.
X
/
B
.
KH
/, called the
Paulsen system [15, Chapter 8] which is defined to be the space of operators
λIKx
y¡µIH:x, y ËX, λ, µ Ëℂ
where
IH
,
IK
denote the identity operators on
H
,
K
respectively. It is well known that [15, Lemma
8.1] any completely contractive map
ϕ
:
X→B
.
H,K
/on the operator space
X
extends canonically to
a unital completely positive (UCP) map S.ϕ/:S.X/→B.KH/defined by
S.ϕ/λIK0x
y¡µIH0/=λIKϕ.x/
ϕ.y/¡µIH.
4 Extreme states and boundary theorems
4.1 Commutant of a TRO and boundary theorem
We introduce the notion of commutant of a rectangular operator set
XB
.
H,K
/for Hilbert spaces
Hand K
. In the context of Hilbert
C¡
-module by Aramba
s
ić [1] introduced a similar notion. We can
now define commutants of operator spaces in general and TROs in particular. We will prove that in
the case of TROs, the commutants satisfy the usual properties with respect to the relevant notions of
invariant subspaces and irreducibility of representations and thereby justifying the term. We observe
that commutant behaves well with respect to the Paulsen map which is crucial for us. Throughout the
artilce unless mentioned otherwise Trepresents a TRO.
Definition 5. For
XB
.
H, K
/, the commutant of
X
is the set in
B
.
KH
/denoted by
Xĺ
and defined
by
Xĺ=ˆA1A2ËB.KH/:A1ËB.K/and A2ËB.H/,A
1x=xA2and
A2x¡=x¡A1,ÅxËX‘
where .A1A2/.η1η2/=A1η1A2η2,η1ËKand η2ËH.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
126
1 INTRODUCTION
Arveson’s extension theorem [3, Theorem1.2.3] for completely positive(CP) maps in the context of
operator systems in
C¡
-algebras is a remarkable result in the study of boundary representations of
C¡
-algebras for operator systems. The theorem asserts that any CP map on an operator system
in
a
C¡
-algebra
to
B
.
/can be extended to a CP map from
to
B
.
/. A natural non-self-adjoint
counterpart of the set up Arveson worked with can be considered to be consisting of operator spaces
in ternary rings of operators(TROs) and completely contractive maps on them. An important recent
work in this scenario is [10] where rectangular matrix convex sets and boundary representations of
operator spaces were introduced. An operator space version of Arveson’s conjecture, namely, every
operator space is completely normed by it’s boundary representations is also established in [10].
Apart from Arveson’s fundamental work [3
–
5], we refer to the work of Douglas Farenick on extremal
theory of matrix states on operator systems [8,9] and Kleski’s work on pure completely positive maps
and boundary representations for operator systems [12].
In this article we study an extension result for rectangular extreme states on operator spaces in
TROs. A characterization result for rectangular extreme state on operator spaces of matrices with
trivial commutants is deduced.
2 Preliminaries
In this section we recall the fundamental notions that we require for the discussions later on in this
article.
Let
H
and
K
be Hilbert spaces and
B
.
H,K
/be the space of all bounded operators from
H
to
K
.
When
K
=
H
, we denote
B
.
H,H
/by
B
.
H
/. A (concrete) operator space is a closed subspace of the
(concrete)
C¡
-algebra
B
.
H
/. The space
B
.
H,K
/can be viewed as a subspace of
B
.
KH
/, and hence it
is always an operator space which we study more in later parts of this article.
An abstract characterisation of operator spaces was established by Ruan [16]. A subclass of operator
spaces called ternary rings of operators referred to as TROs is of special interest to us here. These
were shown to be the injective objects in the category of operator spaces and completely contractive
linear maps by Ruan [17].
Definition 1. A ternary ring of operators (TRO)
T
is a subspace of the
C¡
-algebra
B
.
H
/of all
bounded operators on a Hilbert space
H
that is closed under the triple product .
x, y, z
/
→xy¡z
for all
x, y, z ËT.
A triple morphism between TROs is a linear map that preserves the triple product.
A triple morphism between TROs can be seen as the top-right corner of a *-homomorphism between
the corresponding linking algebras [11]; see also [6, Corollary 8.3.5].
Clearly, an obvious, but important example of a TRO is the space
B
.
H,K
/, where
H
and
K
are
Hilbert spaces.
Definition 2. Let
X
and
Y
be operator spaces. A linear map
ϕ
:
X→Y
is called completely
contractive (CC) if the linear map nϕ:nX→nYis contractive for all n.
We denote the set of all CC maps from Xto B.H,K/by CC.X, B.H,K//.
Definition 3. A representation of a TRO
T
is a triple morphism
ϕ
:
T→B
.
H,K
/for some Hilbert
spaces Hand K.
A representation
ϕ
:
T→B
.
H,K
/is irreducible if, whenever
p
,
q
are projections in
B
.
H
/and
B
.
K
/respectively, such that
qϕ
.
x
/=
ϕ
.
x
/
p
for every
x
Ë
T
, one has
p
=0and
q
=0, or
p
=1and
q=1.
https://doi.org/10.17993/3ctic.2022.112.124-134
A linear map
ϕ
:T
→B
.
H,K
/is nondegenerate if, whenever
p
,
q
are projections in
B
.
H
/and
B.K/, respectively, such that qϕ.x/=ϕ.x/p=0for every xËT, one has p=0and q=0.
Definition 4. A rectangular operator state on an operator space
X
is a non degenerate linear map
ϕ:X→B.H,K/such that ϕcb =1where the norm here is the completely bounded norm.
Several characterizations of nondegenerate and irreducible representations of TROs are obtained
in [7, Lemma 3.1.4 and Lemma 3.1.5].
A concrete TRO T
B
.
H,K
/is said to act irreducibly if the corresponding inclusion representation
is irreducible.
3 systems and spaces
An operator system in a
C¡
-algebra is a unital selfadjoint closed linear subspace. Let
be an operator
system in a C¡-algebra , and be any other C¡-algebra.
A linear map
ϕ
:
→
is called completely positive (CP) if the linear map
nϕ
:
n→n
is
positive for all natural numbers n.
We denote the set of all unital CP maps from to B.H/by UCP.,B.H//.
One of the crucial theorems in this context is Arveson’s extension theorem [3, Theorem1.2.3].
Arveson’s extension theorem asserts that any CP map on an operator system
to
B
.
H
/can be
extended to a CP map from the C¡-algebra to B.H/.
Given an operator space
XB
.
H,K
/, we can assign an operator system
S
.
X
/
B
.
KH
/, called the
Paulsen system [15, Chapter 8] which is defined to be the space of operators
λIKx
y¡µIH:x, y ËX, λ, µ Ëℂ
where
IH
,
IK
denote the identity operators on
H
,
K
respectively. It is well known that [15, Lemma
8.1] any completely contractive map
ϕ
:
X→B
.
H,K
/on the operator space
X
extends canonically to
a unital completely positive (UCP) map S.ϕ/:S.X/→B.KH/defined by
S.ϕ/λIK0x
y¡µIH0/=λIKϕ.x/
ϕ.y/¡µIH.
4 Extreme states and boundary theorems
4.1 Commutant of a TRO and boundary theorem
We introduce the notion of commutant of a rectangular operator set
XB
.
H,K
/for Hilbert spaces
Hand K
. In the context of Hilbert
C¡
-module by Aramba
s
ić [1] introduced a similar notion. We can
now define commutants of operator spaces in general and TROs in particular. We will prove that in
the case of TROs, the commutants satisfy the usual properties with respect to the relevant notions of
invariant subspaces and irreducibility of representations and thereby justifying the term. We observe
that commutant behaves well with respect to the Paulsen map which is crucial for us. Throughout the
artilce unless mentioned otherwise Trepresents a TRO.
Definition 5. For
XB
.
H, K
/, the commutant of
X
is the set in
B
.
KH
/denoted by
Xĺ
and defined
by
Xĺ=ˆA1A2ËB.KH/:A1ËB.K/and A2ËB.H/,A
1x=xA2and
A2x¡=x¡A1,ÅxËX‘
where .A1A2/.η1η2/=A1η1A2η2,η1ËKand η2ËH.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
127
Remark 1. For any non-empty set
XB
.
H,K
/, the commutant
Xĺ
is a von Neumann subalgebra of
B.KH/.
Definition 6. Let
ϕ
:
T→B
.
H,K
/be a non-zero representation and
H1H
and
K1K
be closed
subspaces. We say that the pair .
H1,K1
/of subspaces is
ϕ
-invariant if
ϕ
.
T
/
H1K1
and
ϕ
.
T
/
¡K1H1
.
The following two results follow easily from the definitions above.
Lemma 1. Let
ϕ
:
T→B
.
H, K
/be a non-zero representation. Let
p
and
q
be the orthogonal projections
on closed subspaces
H1H
and
K1K
respectively. Then .
H1,K
1
/is a
ϕ
-invariant pair of subspaces
if and only if qpËϕ.T/ĺ.
Corollary 1. Let
ϕ
:
T→B
.
H, K
/be a representation. Then
ϕ
is irreducible if and only if
ϕ
has no
ϕ-invariant pair of subspaces other than .0,0/ and .H, K/.
The following result illustrate that the term commutant used above is an appropriate one.
Proposition 1. Let
ϕ
:
T→B
.
H, K
/be a non zero representation of a TRO
T
. Then
ϕ
is irreducible
if and only if ϕ.T/ĺ=ℂ.IKIH/.
Proof. Assume
ϕ
is irreducible. Let
qp
be a projection in
ϕ
.
T
/
ĺ
. From the definition of commutant
we have
ϕ
.
x
/
p
=
qϕ
.
x
/. By irreducibility of
ϕ
we have either
p
=0and
q
=0, or
p
=1and
q
=1.
Hence ϕ.T/ĺ=ℂ.IKIH/.
Conversely let
ϕ
.
T
/
ĺ
=
ℂ
.
IKIH
/and
p, q
be projections such that
ϕ
.
x
/
p
=
qϕ
.
x
/. Applying adjoint
we have
pϕ
.
x
/
¡
=
ϕ
.
x
/
¡q
. Thus
qp
Ë
ϕ
.
T
/
ĺ
. By assumption, either
p
=0and
q
=0, or
p
=1and
q
=1.
Hence ϕis irreducible.
The following result shows that commutant behaves well with respect to the Paulsen map.
Lemma 2. For a rectangular matrix state ϕ:X→n,m.ℂ/we have
ϕ.X/ĺ=[S.ϕ/.S.X//]ĺ.
Proof. To show ϕ.X/ĺ[S.ϕ/.S.X//]ĺ, consider A1A2Ëϕ.X/ĺ, then we have
λϕ.x/
ϕ.y/¡µA10
0A2=λA1ϕ.x/A2
ϕ.y/¡A1µA2
=λA1A1ϕ.x/
A2ϕ.y/¡µA2
=A10
0A2 λϕ.x/
ϕ.y/¡µ
Hence A1A2Ë[S.ϕ/.S.X//]ĺ.
Conversely, since the Paulsen system contains a copy of scalar matrices, if
A
Ë[
S
.
ϕ
/.
S
.
X
//]
ĺ
, then
A
=
A1A2
, where
A1
Ë
B
.
K
/and
A2
Ë
B
.
H
/. Using the commutativity relation of the Paulsen map
on the matrices 0ϕ.x/
00
and 00
ϕ.x/¡0
we can conclude that
A1ϕ
.
x
/=
ϕ
.
x
/
A2and A2ϕ
.
x
/
¡
=
ϕ
.
x
/
¡A1,
Å
x
Ë
X
. Hence
A
=
A1A2
Ë
ϕ
.
X
/
ĺ
.
Thus [S.ϕ/.S.X//]ĺϕ.X/ĺ.
https://doi.org/10.17993/3ctic.2022.112.124-134
We prove a version of the rectangular boundary theorem proved in [10, Theorem 2.17] which is an
analogue of boundary theorem of Arveson in the context of operator systems. Arveson’s boundary
theorem [4, Theorem 2.1.1] states that if
B
.
H
/is an operator system which acts irreducibly
on
H
such that the
C¡
-algebra
C¡
.
/contains the algebra of compact operators
.
H
/, then the
identity representation of
C¡
.
/is a boundary representation for
if and only if the quotient map
B
.
H
/
→B
.
H
/˙
.
H
/is not completely isometric on
. Our context here is that of an operator space
and the generated TRO.
Throughout we assume that
X
is an operator space, and
T
is a TRO containing
X
as a generating
set. We say that a rectangular operator state
ϕ
:
X→B
.
H,K
/has unique extension property if any
rectangular operator state
ϕ
on
T
whose restriction to
X
coincides with
ϕ
is automatically a triple
morphism. Boundary representations for operator spaces were introduced in [10]. For our purposes we
consider the following definition which is slightly different from the definition given in [10, Definition
2.7] but we remark that both are the same.
Definition 7. An irreducible representation
θ
:
T→B
.
H,K
/is a boundary representation for
X
if
θXis a rectangular operator state on Xand θXhas unique extension property.
An exact sequence of TROs induces an exact sequence of the corresponding linking algebras, which
is actually an exact sequence of
C¡
-algebras. The decomposition result for representations of
C¡
-algebras
is well known [5, section 7] and the result for TROs follows from it.
If 0
→T0→T→Tĺ→
0is an exact sequence of TRO’s, then every non degenerate representation
π:T→B.H,K/
of
T
decomposes uniquely into a direct sum of representations
π
=
π0
Tπĺ
T
where
π0
T
is the unique
extension to
T
of a nondegenerate representation of the TRO-ideal [7, Definition 2.2.7]
T0
, and where
πĺ
T
is a nondegenerate representation of
T
that annihilates
T0
. When
π
=
π0
T
we say that
π
lives on
T0
.
With
X
and
T
as above let
.
H,K
/denotes the set of compact operators from
H
to
K
. Then
T
=
T
ã
.
H,K
/is a TRO-ideal and we have an obvious exact sequence of TRO’s. Denote
T
to
be the set of all irreducible triple morphisms
θ
:
T→B
.
H,K
/such that
θ
lives on
T
and
θX
is a
rectangular operator state on X. The following is a boundary theorem in this context.
Theorem 1. Let
XB
.
H,K
/be an operator space such that TRO
T
generated by
X
acts irreducibly
and
T≠
ˆ0‘. Then
T
contains a boundary representation for
X
if and only if the quotient map
q:T→T˙Tis not completely isometric on X.
Proof. Assume that the quotient map
q
is not completely isometric on
X
. If
T
contains no boundary
representations for
X
, then every boundary representation must annihilate
T
and consequently it
factors through
q
. By [10, Theorem 2.9], there are sufficiently many boundary representations
πl
,
l
Ë
I
,
for Xso that
.q.xij /≤.xij / = sup
lËI
πl.xij /≤.q.xij /
for every nİnmatrix .xij /over Xand every n≥1. This is a contradiction.
Conversely, if the quotient map is completely isometric on
X
, then we show that no
π
Ë
T
can be
a boundary representation for
X
. Let
π
:
T→B
.
H,K
/be an irreducible representation that lives on
T
and consider the map
Q
:
q
.
T
/
→B
.
KH
/defined by
Q
.
q
.
a
// =
π
.
a
/. Then
Q
is well defined and
Q.q.a// = π.a/≤a=q.a/.
Similarly
Qn
.
q
.
aij
//
≤
.
q
.
aij
// and hence
Q
is completely contractive. Then the Paulsen’s map
S
.
Q
/:
S
.
q
.
T
//
→B
.
KH
/is unital and completely positive. By Arveson’s extension theorem
S
.
Q
/
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
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128
Remark 1. For any non-empty set
XB
.
H,K
/, the commutant
Xĺ
is a von Neumann subalgebra of
B.KH/.
Definition 6. Let
ϕ
:
T→B
.
H,K
/be a non-zero representation and
H1H
and
K1K
be closed
subspaces. We say that the pair .
H1,K1
/of subspaces is
ϕ
-invariant if
ϕ
.
T
/
H1K1
and
ϕ
.
T
/
¡K1H1
.
The following two results follow easily from the definitions above.
Lemma 1. Let
ϕ
:
T→B
.
H, K
/be a non-zero representation. Let
p
and
q
be the orthogonal projections
on closed subspaces
H1H
and
K1K
respectively. Then .
H1,K
1
/is a
ϕ
-invariant pair of subspaces
if and only if qpËϕ.T/ĺ.
Corollary 1. Let
ϕ
:
T→B
.
H, K
/be a representation. Then
ϕ
is irreducible if and only if
ϕ
has no
ϕ-invariant pair of subspaces other than .0,0/ and .H, K/.
The following result illustrate that the term commutant used above is an appropriate one.
Proposition 1. Let
ϕ
:
T→B
.
H, K
/be a non zero representation of a TRO
T
. Then
ϕ
is irreducible
if and only if ϕ.T/ĺ=ℂ.IKIH/.
Proof. Assume
ϕ
is irreducible. Let
qp
be a projection in
ϕ
.
T
/
ĺ
. From the definition of commutant
we have
ϕ
.
x
/
p
=
qϕ
.
x
/. By irreducibility of
ϕ
we have either
p
=0and
q
=0, or
p
=1and
q
=1.
Hence ϕ.T/ĺ=ℂ.IKIH/.
Conversely let
ϕ
.
T
/
ĺ
=
ℂ
.
IKIH
/and
p, q
be projections such that
ϕ
.
x
/
p
=
qϕ
.
x
/. Applying adjoint
we have
pϕ
.
x
/
¡
=
ϕ
.
x
/
¡q
. Thus
qp
Ë
ϕ
.
T
/
ĺ
. By assumption, either
p
=0and
q
=0, or
p
=1and
q
=1.
Hence ϕis irreducible.
The following result shows that commutant behaves well with respect to the Paulsen map.
Lemma 2. For a rectangular matrix state ϕ:X→n,m.ℂ/we have
ϕ.X/ĺ=[S.ϕ/.S.X//]ĺ.
Proof. To show ϕ.X/ĺ[S.ϕ/.S.X//]ĺ, consider A1A2Ëϕ.X/ĺ, then we have
λϕ.x/
ϕ.y/¡µA10
0A2=λA1ϕ.x/A2
ϕ.y/¡A1µA2
=λA1A1ϕ.x/
A2ϕ.y/¡µA2
=A10
0A2 λϕ.x/
ϕ.y/¡µ
Hence A1A2Ë[S.ϕ/.S.X//]ĺ.
Conversely, since the Paulsen system contains a copy of scalar matrices, if
A
Ë[
S
.
ϕ
/.
S
.
X
//]
ĺ
, then
A
=
A1A2
, where
A1
Ë
B
.
K
/and
A2
Ë
B
.
H
/. Using the commutativity relation of the Paulsen map
on the matrices 0ϕ.x/
00
and 00
ϕ.x/¡0
we can conclude that
A1ϕ
.
x
/=
ϕ
.
x
/
A2and A2ϕ
.
x
/
¡
=
ϕ
.
x
/
¡A1,
Å
x
Ë
X
. Hence
A
=
A1A2
Ë
ϕ
.
X
/
ĺ
.
Thus [S.ϕ/.S.X//]ĺϕ.X/ĺ.
https://doi.org/10.17993/3ctic.2022.112.124-134
We prove a version of the rectangular boundary theorem proved in [10, Theorem 2.17] which is an
analogue of boundary theorem of Arveson in the context of operator systems. Arveson’s boundary
theorem [4, Theorem 2.1.1] states that if
B
.
H
/is an operator system which acts irreducibly
on
H
such that the
C¡
-algebra
C¡
.
/contains the algebra of compact operators
.
H
/, then the
identity representation of
C¡
.
/is a boundary representation for
if and only if the quotient map
B
.
H
/
→B
.
H
/˙
.
H
/is not completely isometric on
. Our context here is that of an operator space
and the generated TRO.
Throughout we assume that
X
is an operator space, and
T
is a TRO containing
X
as a generating
set. We say that a rectangular operator state
ϕ
:
X→B
.
H,K
/has unique extension property if any
rectangular operator state
ϕ
on
T
whose restriction to
X
coincides with
ϕ
is automatically a triple
morphism. Boundary representations for operator spaces were introduced in [10]. For our purposes we
consider the following definition which is slightly different from the definition given in [10, Definition
2.7] but we remark that both are the same.
Definition 7. An irreducible representation
θ
:
T→B
.
H,K
/is a boundary representation for
X
if
θXis a rectangular operator state on Xand θXhas unique extension property.
An exact sequence of TROs induces an exact sequence of the corresponding linking algebras, which
is actually an exact sequence of
C¡
-algebras. The decomposition result for representations of
C¡
-algebras
is well known [5, section 7] and the result for TROs follows from it.
If 0
→T0→T→Tĺ→
0is an exact sequence of TRO’s, then every non degenerate representation
π:T→B.H,K/
of
T
decomposes uniquely into a direct sum of representations
π
=
π0
Tπĺ
T
where
π0
T
is the unique
extension to
T
of a nondegenerate representation of the TRO-ideal [7, Definition 2.2.7]
T0
, and where
πĺ
T
is a nondegenerate representation of
T
that annihilates
T0
. When
π
=
π0
T
we say that
π
lives on
T0
.
With
X
and
T
as above let
.
H,K
/denotes the set of compact operators from
H
to
K
. Then
T
=
T
ã
.
H,K
/is a TRO-ideal and we have an obvious exact sequence of TRO’s. Denote
T
to
be the set of all irreducible triple morphisms
θ
:
T→B
.
H,K
/such that
θ
lives on
T
and
θX
is a
rectangular operator state on X. The following is a boundary theorem in this context.
Theorem 1. Let
XB
.
H,K
/be an operator space such that TRO
T
generated by
X
acts irreducibly
and
T≠
ˆ0‘. Then
T
contains a boundary representation for
X
if and only if the quotient map
q:T→T˙Tis not completely isometric on X.
Proof. Assume that the quotient map
q
is not completely isometric on
X
. If
T
contains no boundary
representations for
X
, then every boundary representation must annihilate
T
and consequently it
factors through
q
. By [10, Theorem 2.9], there are sufficiently many boundary representations
πl
,
l
Ë
I
,
for Xso that
.q.xij /≤.xij / = sup
lËI
πl.xij /≤.q.xij /
for every nİnmatrix .xij /over Xand every n≥1. This is a contradiction.
Conversely, if the quotient map is completely isometric on
X
, then we show that no
π
Ë
T
can be
a boundary representation for
X
. Let
π
:
T→B
.
H,K
/be an irreducible representation that lives on
T
and consider the map
Q
:
q
.
T
/
→B
.
KH
/defined by
Q
.
q
.
a
// =
π
.
a
/. Then
Q
is well defined and
Q.q.a// = π.a/≤a=q.a/.
Similarly
Qn
.
q
.
aij
//
≤
.
q
.
aij
// and hence
Q
is completely contractive. Then the Paulsen’s map
S
.
Q
/:
S
.
q
.
T
//
→B
.
KH
/is unital and completely positive. By Arveson’s extension theorem
S
.
Q
/
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
129
extends to a completely positive map S.Q/:C¡.S.q.T/// →B.KH/. Define ψ:T→B.H,K/via
S.Q/0q.a/
00
=¡ψ.a/
¡¡
Then clearly ψis linear. Also for all aËT,
ψ.a/=0ψ.a/
00
≤S.Q/0q.a/
00
 ≤S.Q/q.a/≤a.
Since
π
Ë
T
and
ψX
=
πX
we have
ψcb
=1. Also we have
ψ
.
T
/=0and hence
π
.
T
/=0which is
a contradiction to the fact that πlives in T.
Remark 2. The following result is a specific form of rectangular boundary theorem given above. Here
we give an independent proof which directly uses the corresponding version of Arveson’s boundary
theorem and Lemma 2.
Theorem 2. Assume that
X
is an operator space in
n,m
.
ℂ
/such that
dim
.
Xĺ
/=1. If
ϕ
:
X→
n,m
.
ℂ
/is a rectangular matrix state on
n,m
.
ℂ
/for which
ϕ
.
x
/=
x
,Å
x
Ë
X
, then
ϕ
.
a
/=
a
,
ÅaËn,m.ℂ/.
Proof. By Lemma 2,
ϕ
.
X
/
ĺ
=[
S
.
ϕ
/.
S
.
X
//]
ĺ
. Hence
dim
[
S
.
ϕ
/.
S
.
X
//]
ĺ
=1. Since
ϕ
.
x
/=
x,
Å
x
Ë
X
,
we have
S
.
ϕ
/.
y
/=
y,
Å
y
Ë
S
.
X
/. Then by [9, Thoerem 4.2] ,
S
.
ϕ
/.
z
/=
z,
Å
z
Ë
n+m
. In
particular,
0ϕ.a/
00
=S.ϕ/0a
00
/=0a
00
,
and hence ϕ.a/=a, ÅaËn,m.ℂ/.
4.2 Rectangular operator extreme states
We prove an important extension result in this section. In this section we prove that any rectangular
operator extreme state on an operator space in a TRO can be extended to a rectangular operator
extreme state on the TRO. Extension results in the same spirit concerning operator systems and UCP
maps were proved by Kleski [12]. The following definition which appeared in [10, Definition 2.9] is
important for our further discussion.
We begin by defining rectangular operator convex combination, and rectangular operator extreme
states.
Definition 8. Suppose that
X
is an operator space, and
ϕ
:
X→B
.
H,K
/is a completely contractive
linear map. A rectangular operator convex combination is an expression
ϕ
=
α¡
1ϕ1β1
+
▽
+
α¡
nϕnβn
,
where
βi
:
H→H⟩
and
αi
:
K→K⟩
are linear maps, and
ϕi
:
X→B
.
H⟩,K⟩
/are completely contractive
linear maps for
i
=1
,
2
,▽n
such that
α¡
1α1
+
▽
+
α¡
nαn
=1and
β¡
1β1
+
▽
+
β¡
nβn
=1. Such a
rectangular convex combination is proper if
αi
,
βi
are surjective, and trivial if
α¡
iαi
=
λi
1,
β¡
iβi
=
λi
1,
and α¡
iϕiβi=λiϕifor some λiË [0,1].
A completely contractive map
ϕ
:
X→B
.
H,K
/is a rectangular operator extreme state if any proper
rectangular operator convex combination ϕ=α¡
1ϕ1β1+▽+α¡
nϕnβnis trivial.
Rectangular operator states will be referred to as rectangular matrix states if the underlying Hilbert
spaces are finite dimensional. The following theorem illustrates a relation between linear extreme states
and and rectangular operator extreme states.
Theorem 3. Let
X1X2
be operator spaces. If a completely contractive map
ϕ
:
X2→B
.
H,K
/is
linear extreme in the set
CC
.
X2,B
.
H,K
// of all completely contractive maps from
X2
to
B
.
H,K
/,
and ϕX1is rectangular operator extreme, then ϕis a rectangular operator extreme state.
https://doi.org/10.17993/3ctic.2022.112.124-134
Proof. Let
ϕ
:
X2→B
.
H,K
/be linear extreme and
ϕX1
be rectangular operator extreme. Then
S
.
ϕ
/:
S
.
X2
/
→B
.
KH
/is linear extreme in
UCP
.
S
.
X2
/,
B
.
KH
//. For, let
ψ1,ψ
2
Ë
UCP
.
S
.
X2
/
,B
.
KH
//
and 0<t<1be such that
S.ϕ/=tΨ1+.1*t/Ψ2.
Then
S
.
ϕ
/*
t
Ψ
1
is UCP, so by ( [10], Lemma 1.11) there exists a completely contractive map
ϕ1:X2→B.H,K/such that
tΨ1λx
y¡µ=w1˙2 λIKϕ1.x/
ϕ1.y/¡µIHw1˙2,
where
w=tΨ110
01
=tIK0
0IH.
So
tΨ1λx
y¡µ=tS.ϕ1/λx
y¡µ.
Thus Ψ1=S.ϕ1/.Similarly Ψ2=S.ϕ2/for some cc map ϕ2:X2→B.H,K/. Hence
S.ϕ/=tS.ϕ1/+.1*t/S.ϕ2/and therefore
ϕ=tϕ1+.1*t/ϕ2.
Since ϕis a linear extreme point, we have ϕ=ϕ1=ϕ2. Thus
S.ϕ/=S.ϕ1/=S.ϕ2/
S.ϕ/=Ψ
1=Ψ
2.
Thus
S
.
ϕ
/:
S
.
X2
/
→B
.
KH
/is a linear extreme point. Since
S
.
ϕ
/
S.X1/
is pure [10, Proposition 1.12],
by [12, Proposition 2.2]
S
.
ϕ
/is a pure UCP map on
S
.
X2
/. Hence
ϕ
is a rectangular operator extreme
state on X2.
Here we consider the set of bounded operators from
X
to
B
.
H,K
/with the weak
¡
topology called
the bounded weak topology or BW-topology, identifying this set with a dual Banach space. In its
relative BW-topology,
CC
.
X, B
.
H,K
// is compact (see [3, Section 1.1] or [15, Chapter 7] for details).
Theorem 4. Let
X1X2
be operator spaces. Then every rectangular operator extreme state on
X1
has
an extension to a rectangular operator extreme state on X2.
Proof. Let ϕËCC.X1,B.H,K// be a rectangular operator extreme state and let
=ˆψËCC.X2,B.H,K// : ψX1=ϕ‘.
Then clearly
is linear convex, and BW-compact. We claim that it is a face. For let
ψ1,ψ
2
Ë
CC.X1,B.H,K// and 0<t<1be such that tψ1+.1*t/ψ2Ë.
Then
tψ1X1+.1*t/ψ2X1=ϕ
tS.ψ1/.S.X1/+.1*t/S.ψ2/S.X1/=S.ϕ/.
Since S.ϕ/is pure,
S.ψ1/S.X1/=S.ψ2/S.X2/=S.ϕ/.
ψ1X1=ψ2X2=ϕ.
⇒ψ1,ψ
2
Ë
and hence
is a face. Thus
has a linear extreme point say
ϕĺ
which is a linear extreme
point of
CC
.
X2,B
.
H,K
//. By Theorem 3, it follows that
ϕĺ
:
X2→B
.
H,K
/is a rectangular operator
extreme state.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
130
extends to a completely positive map S.Q/:C¡.S.q.T/// →B.KH/. Define ψ:T→B.H,K/via
S.Q/0q.a/
00
=¡ψ.a/
¡¡
Then clearly ψis linear. Also for all aËT,
ψ.a/=0ψ.a/
00
≤S.Q/0q.a/
00
 ≤S.Q/q.a/≤a.
Since
π
Ë
T
and
ψX
=
πX
we have
ψcb
=1. Also we have
ψ
.
T
/=0and hence
π
.
T
/=0which is
a contradiction to the fact that πlives in T.
Remark 2. The following result is a specific form of rectangular boundary theorem given above. Here
we give an independent proof which directly uses the corresponding version of Arveson’s boundary
theorem and Lemma 2.
Theorem 2. Assume that
X
is an operator space in
n,m
.
ℂ
/such that
dim
.
Xĺ
/=1. If
ϕ
:
X→
n,m
.
ℂ
/is a rectangular matrix state on
n,m
.
ℂ
/for which
ϕ
.
x
/=
x
,Å
x
Ë
X
, then
ϕ
.
a
/=
a
,
ÅaËn,m.ℂ/.
Proof. By Lemma 2,
ϕ
.
X
/
ĺ
=[
S
.
ϕ
/.
S
.
X
//]
ĺ
. Hence
dim
[
S
.
ϕ
/.
S
.
X
//]
ĺ
=1. Since
ϕ
.
x
/=
x,
Å
x
Ë
X
,
we have
S
.
ϕ
/.
y
/=
y,
Å
y
Ë
S
.
X
/. Then by [9, Thoerem 4.2] ,
S
.
ϕ
/.
z
/=
z,
Å
z
Ë
n+m
. In
particular,
0ϕ.a/
00
=S.ϕ/0a
00
/=0a
00
,
and hence ϕ.a/=a, ÅaËn,m.ℂ/.
4.2 Rectangular operator extreme states
We prove an important extension result in this section. In this section we prove that any rectangular
operator extreme state on an operator space in a TRO can be extended to a rectangular operator
extreme state on the TRO. Extension results in the same spirit concerning operator systems and UCP
maps were proved by Kleski [12]. The following definition which appeared in [10, Definition 2.9] is
important for our further discussion.
We begin by defining rectangular operator convex combination, and rectangular operator extreme
states.
Definition 8. Suppose that
X
is an operator space, and
ϕ
:
X→B
.
H,K
/is a completely contractive
linear map. A rectangular operator convex combination is an expression
ϕ
=
α¡
1ϕ1β1
+
▽
+
α¡
nϕnβn
,
where
βi
:
H→H⟩
and
αi
:
K→K⟩
are linear maps, and
ϕi
:
X→B
.
H⟩,K⟩
/are completely contractive
linear maps for
i
=1
,
2
,▽n
such that
α¡
1α1
+
▽
+
α¡
nαn
=1and
β¡
1β1
+
▽
+
β¡
nβn
=1. Such a
rectangular convex combination is proper if
αi
,
βi
are surjective, and trivial if
α¡
iαi
=
λi
1,
β¡
iβi
=
λi
1,
and α¡
iϕiβi=λiϕifor some λiË [0,1].
A completely contractive map
ϕ
:
X→B
.
H,K
/is a rectangular operator extreme state if any proper
rectangular operator convex combination ϕ=α¡
1ϕ1β1+▽+α¡
nϕnβnis trivial.
Rectangular operator states will be referred to as rectangular matrix states if the underlying Hilbert
spaces are finite dimensional. The following theorem illustrates a relation between linear extreme states
and and rectangular operator extreme states.
Theorem 3. Let
X1X2
be operator spaces. If a completely contractive map
ϕ
:
X2→B
.
H,K
/is
linear extreme in the set
CC
.
X2,B
.
H,K
// of all completely contractive maps from
X2
to
B
.
H,K
/,
and ϕX1is rectangular operator extreme, then ϕis a rectangular operator extreme state.
https://doi.org/10.17993/3ctic.2022.112.124-134
Proof. Let
ϕ
:
X2→B
.
H,K
/be linear extreme and
ϕX1
be rectangular operator extreme. Then
S
.
ϕ
/:
S
.
X2
/
→B
.
KH
/is linear extreme in
UCP
.
S
.
X2
/,
B
.
KH
//. For, let
ψ1,ψ
2
Ë
UCP
.
S
.
X2
/
,B
.
KH
//
and 0<t<1be such that
S.ϕ/=tΨ1+.1*t/Ψ2.
Then
S
.
ϕ
/*
t
Ψ
1
is UCP, so by ( [10], Lemma 1.11) there exists a completely contractive map
ϕ1:X2→B.H,K/such that
tΨ1λx
y¡µ=w1˙2 λIKϕ1.x/
ϕ1.y/¡µIHw1˙2,
where
w=tΨ110
01
=tIK0
0IH.
So
tΨ1λx
y¡µ=tS.ϕ1/λx
y¡µ.
Thus Ψ1=S.ϕ1/.Similarly Ψ2=S.ϕ2/for some cc map ϕ2:X2→B.H,K/. Hence
S.ϕ/=tS.ϕ1/+.1*t/S.ϕ2/and therefore
ϕ=tϕ1+.1*t/ϕ2.
Since ϕis a linear extreme point, we have ϕ=ϕ1=ϕ2. Thus
S.ϕ/=S.ϕ1/=S.ϕ2/
S.ϕ/=Ψ
1=Ψ
2.
Thus
S
.
ϕ
/:
S
.
X2
/
→B
.
KH
/is a linear extreme point. Since
S
.
ϕ
/
S.X1/
is pure [10, Proposition 1.12],
by [12, Proposition 2.2]
S
.
ϕ
/is a pure UCP map on
S
.
X2
/. Hence
ϕ
is a rectangular operator extreme
state on X2.
Here we consider the set of bounded operators from
X
to
B
.
H,K
/with the weak
¡
topology called
the bounded weak topology or BW-topology, identifying this set with a dual Banach space. In its
relative BW-topology,
CC
.
X, B
.
H,K
// is compact (see [3, Section 1.1] or [15, Chapter 7] for details).
Theorem 4. Let
X1X2
be operator spaces. Then every rectangular operator extreme state on
X1
has
an extension to a rectangular operator extreme state on X2.
Proof. Let ϕËCC.X1,B.H,K// be a rectangular operator extreme state and let
=ˆψËCC.X2,B.H,K// : ψX1=ϕ‘.
Then clearly
is linear convex, and BW-compact. We claim that it is a face. For let
ψ1,ψ
2
Ë
CC.X1,B.H,K// and 0<t<1be such that tψ1+.1*t/ψ2Ë.
Then
tψ1X1+.1*t/ψ2X1=ϕ
tS.ψ1/.S.X1/+.1*t/S.ψ2/S.X1/=S.ϕ/.
Since S.ϕ/is pure,
S.ψ1/S.X1/=S.ψ2/S.X2/=S.ϕ/.
ψ1X1=ψ2X2=ϕ.
⇒ψ1,ψ
2
Ë
and hence
is a face. Thus
has a linear extreme point say
ϕĺ
which is a linear extreme
point of
CC
.
X2,B
.
H,K
//. By Theorem 3, it follows that
ϕĺ
:
X2→B
.
H,K
/is a rectangular operator
extreme state.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
131
Now, in view of the above results, extension of rectangular operator extreme state from an operator
space to the generated TRO is immediate.
Corollary 2. If
X
is an operator space and
T
is the TRO generated by
X
and containing
X
, then any
rectangular operator extreme state on the operator space
X
can be extended to a rectangular operator
extreme state on the TRO T.
Proof. Follows directly from the Theorem 4 by taking X1=Xand X2=T.
4.3 Rectangular matrix extreme states
Here we take up the special case of finite dimensional rectangular operator states and show that they are
isometric or unitary ‘conjugates’ of the identity map. Here we investigate the relation between matrix
extreme states on operator spaces and commutants of images of operator spaces under rectangular
matrix extreme states. For operator spaces in rectangular matrix algebras with trivial commutants, we
deduce that rectangular matrix extreme states are certain ’conjugates’ of the identity state.
Proposition 2. If
ϕ
:
X→n,m
.
ℂ
/is a rectangular matrix extreme state on the operator space
X
,
then dim.ϕ.X/ĺ/=1.
Proof. The commutant ϕ.X/ĺis a unital ¡-subalgebra of n+m.ℂ/and is therefore the linear span
of its projections. Choose any nonzero projection
pq
Ë
ϕ
.
X
/
ĺ
. Then
ϕ
=
qϕp
+.
I
*
q
/
ϕ
.
I
*
p
/
.
Since
ϕ
is a rectangular matrix extreme point, we have
p¡p
=
λ1I
,
q¡q
=
λ1I
and .
I
*
p
/
¡
.
I
*
p
/=
λ2I
,
.
I
*
q
/
¡
.
I
*
q
/=
λ2I
and
qϕp
=
λ1ϕ
,.
I
*
q
/
ϕ
.
I
*
p
/=
λ2ϕ
, for some
λ1,λ
2
Ë [0
,
1]. Thus
λ2
1
=
λ1
and
λ2
2=λ2. This gives p=Iand q=I. Therefore ϕ.X/ĺ=ℂI. Hence dim.ϕ.X/ĺ/=1.
Theorem 5. Assume that
X
is an operator space in
n,m
.
ℂ
/and that
dim.Xĺ/=1.
1.
If
ϕ
:
X→r,s
.
ℂ
/is a rectangular matrix extreme state on
X
, then
r≤n
,
s≤m
and there are
isometries w:ℂr→ℂnand v:ℂs→ℂmsuch that ϕ.x/=w¡xv,ÅxËX.
2.
A rectangular matrix state
ϕ
on
X
with values in
n,m
.
ℂ
/is rectangular matrix extreme if and
only if there exist unitaries vËn.ℂ/and uËm.ℂ/such that ϕ.x/=v¡xu, ÅxËX.
Proof. (1): Let
Xn,m
.
ℂ
/, and
dim
.
Xĺ
/=1. Let
ϕ
:
X→r,s
.
ℂ
/be a rectangular matrix extreme
state. By Corollary 2,
ϕ
can be extended to a rectangular extreme state Φ:
n,m
.
ℂ
/
→r,s
.
ℂ
/. Then
S
.Φ/ :
S
.
n,m
.
ℂ
//
→B
.
ℂrℂs
/is pure. So by [12, Theorem 3.3] there exists a boundary representation
w:n+m.ℂ// →B.L/for S.n,m.ℂ// and an isometry u:ℂrℂs→Lsuch that
S.ϕ/.y/=u¡w.y/u, for all yËS.n,m.ℂ//.
By [10, Proposition 2.8] we can decompose
L
as an orthogonal direct sum
L
=
KwHw
in such a way
that w=S.π/for some irreducible representation π:n,m.ℂ/→B.Hw,Kw/.
From the construction of
Hwand Kw
, it follows that
u
maps
ℂr
0to
Kw
and 0
ℂs
to
Hw
. By defining
the maps
u1
and
u2
as
u1
.
x
/=
u
.
x
0/,
x
Ë
ℂr
and
u1
.
y
/=
u
.0
y
/,
y
Ë
ℂs
we see that
u1
:
ℂr→Kw
and
u2
:
ℂs→Hw
are isometries such that
u
=
u1u2
. Then Φ.
x
/=
u¡
1π
.
x
/
u2
,Å
x
Ë
n,m
.
ℂ
/and thus
Φis a compression of an irreducible representation of
n,m
.
ℂ
/. Since every irreducible representation
of
n,m
.
ℂ
/is unitarily equivalent to the identity representation [7, Lemma 3.2.3] we have that
ϕ
is a
compression of the identity representation. That is, there are isometries
w
:
ℂr→ℂn
and
v
:
ℂs→ℂm
such that
ϕ.x/=w¡xv, ÅxËX.
Since vand ware isometries, we conclude that r≤nand s≤m.
https://doi.org/10.17993/3ctic.2022.112.124-134
(2) Let
ϕ
:
X→n,m
.
ℂ
/be a rectangular matrix extreme state. Then by part .
a
/, there are
isometries (unitaries in this case) uËn.ℂ/,vËm.ℂ/such that
ϕ.x/=v¡xu, ÅxËX.
Conversely let ϕ.x/=v¡xu,ÅxËXfor unitaries uand vthen
S.ϕ/λx
y¡µ=u¡0
0v¡λx
y¡µu0
0v,Åx, y ËX.
Then by [9, Theorem 4.3]
S
.
ϕ
/is pure. Hence
ϕ
is a rectangular matrix extreme state by ( [10,
Proposition2.12]).
The following result is now an immediate consequence of Proposition 2 and Theorem 5.
Theorem 6. Let
Xn,m
.
ℂ
/be an operator space. Then the inclusion map
i
.
x
/=
x,
Å
x
Ë
X
,isa
rectangular matrix extreme state if and only if dim.Xĺ/=1.
REFERENCES
[1]
Aramba
s
ić L.(2005). Irreducible representations of Hilbert C*-modules. Math. Proc. Roy. Irish
Acad., 105 A, 11-14; MR2162903.
[2]
Arunkumar C.S.,Shabna A. M.,Syamkrishnan M. S. and Vijayarajan A. K. (2021).
Extreme states on operator spaces in ternary rings of operators. Proc. Ind. Acad. Sci.131, 44, MR
4338047.
[3] Arveson W. B. (1969). Subalgebras of C¡-algebras. Acta Math. 123, 141-224; MR0253059.
[4]
Arveson W. B.(1972). Subalgebras of
C¡
-algebras
II
.Acta Math. 128(1972) no. 3-4 , 271-308;
MR0394232.
[5]
Arveson W. B.(2011). The noncommutative Choquet boundary II: Hyperrigidity. Israel J. Math.
184 (2011), 349-385; MR2823981.
[6]
Blecher D. P. and Christian Le Merdy (2004). Operator algebras and their modules-an operator
space approach. London Mathematical Society Monographs, New Series, vol. 30, Oxford University
Press, Oxford.
[7]
Bohle D. (2011). K-Theory for ternary structures, Ph.D Thesis, Welstfälishe Wilhelms-Universität
Münster.
[8]
Farenick D. R. (2000). Extremal Matrix states on operator Systems. Journal of London Mathe-
matical Society 61, no. 3, 885-892; MR1766112.
[9]
Farenick D. R.(2004). Pure matrix states on operator systems. Linear Algebra and its Applications
393, 149-173; MR2098611.
[10]
Fuller A. H.,Hartz M. and Lupini M. (2018). Boundary representations of operator spaces,
and compact rectangular matrix convex sets. Journal of Operator Theory, Vol.79, No.1, 139-172;
MR3764146.
[11]
Hamana M. (1999). Triple envelopes and Shilov boundaries of operator spaces. Mathematical
Journal of Toyama University 22, 77-93; MR1744498.
[12]
Kleski C. (2014). Boundary representations and pure completely positive maps. Journal of
Operator Theory, pages 45-62; MR3173052.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
132
Now, in view of the above results, extension of rectangular operator extreme state from an operator
space to the generated TRO is immediate.
Corollary 2. If
X
is an operator space and
T
is the TRO generated by
X
and containing
X
, then any
rectangular operator extreme state on the operator space
X
can be extended to a rectangular operator
extreme state on the TRO T.
Proof. Follows directly from the Theorem 4 by taking X1=Xand X2=T.
4.3 Rectangular matrix extreme states
Here we take up the special case of finite dimensional rectangular operator states and show that they are
isometric or unitary ‘conjugates’ of the identity map. Here we investigate the relation between matrix
extreme states on operator spaces and commutants of images of operator spaces under rectangular
matrix extreme states. For operator spaces in rectangular matrix algebras with trivial commutants, we
deduce that rectangular matrix extreme states are certain ’conjugates’ of the identity state.
Proposition 2. If
ϕ
:
X→n,m
.
ℂ
/is a rectangular matrix extreme state on the operator space
X
,
then dim.ϕ.X/ĺ/=1.
Proof. The commutant ϕ.X/ĺis a unital ¡-subalgebra of n+m.ℂ/and is therefore the linear span
of its projections. Choose any nonzero projection
pq
Ë
ϕ
.
X
/
ĺ
. Then
ϕ
=
qϕp
+.
I
*
q
/
ϕ
.
I
*
p
/
.
Since
ϕ
is a rectangular matrix extreme point, we have
p¡p
=
λ1I
,
q¡q
=
λ1I
and .
I
*
p
/
¡
.
I
*
p
/=
λ2I
,
.
I
*
q
/
¡
.
I
*
q
/=
λ2I
and
qϕp
=
λ1ϕ
,.
I
*
q
/
ϕ
.
I
*
p
/=
λ2ϕ
, for some
λ1,λ
2
Ë [0
,
1]. Thus
λ2
1
=
λ1
and
λ2
2=λ2. This gives p=Iand q=I. Therefore ϕ.X/ĺ=ℂI. Hence dim.ϕ.X/ĺ/=1.
Theorem 5. Assume that
X
is an operator space in
n,m
.
ℂ
/and that
dim.Xĺ/=1.
1.
If
ϕ
:
X→r,s
.
ℂ
/is a rectangular matrix extreme state on
X
, then
r≤n
,
s≤m
and there are
isometries w:ℂr→ℂnand v:ℂs→ℂmsuch that ϕ.x/=w¡xv,ÅxËX.
2.
A rectangular matrix state
ϕ
on
X
with values in
n,m
.
ℂ
/is rectangular matrix extreme if and
only if there exist unitaries vËn.ℂ/and uËm.ℂ/such that ϕ.x/=v¡xu, ÅxËX.
Proof. (1): Let
Xn,m
.
ℂ
/, and
dim
.
Xĺ
/=1. Let
ϕ
:
X→r,s
.
ℂ
/be a rectangular matrix extreme
state. By Corollary 2,
ϕ
can be extended to a rectangular extreme state Φ:
n,m
.
ℂ
/
→r,s
.
ℂ
/. Then
S
.Φ/ :
S
.
n,m
.
ℂ
//
→B
.
ℂrℂs
/is pure. So by [12, Theorem 3.3] there exists a boundary representation
w:n+m.ℂ// →B.L/for S.n,m.ℂ// and an isometry u:ℂrℂs→Lsuch that
S.ϕ/.y/=u¡w.y/u, for all yËS.n,m.ℂ//.
By [10, Proposition 2.8] we can decompose
L
as an orthogonal direct sum
L
=
KwHw
in such a way
that w=S.π/for some irreducible representation π:n,m.ℂ/→B.Hw,Kw/.
From the construction of
Hwand Kw
, it follows that
u
maps
ℂr
0to
Kw
and 0
ℂs
to
Hw
. By defining
the maps
u1
and
u2
as
u1
.
x
/=
u
.
x
0/,
x
Ë
ℂr
and
u1
.
y
/=
u
.0
y
/,
y
Ë
ℂs
we see that
u1
:
ℂr→Kw
and
u2
:
ℂs→Hw
are isometries such that
u
=
u1u2
. Then Φ.
x
/=
u¡
1π
.
x
/
u2
,Å
x
Ë
n,m
.
ℂ
/and thus
Φis a compression of an irreducible representation of
n,m
.
ℂ
/. Since every irreducible representation
of
n,m
.
ℂ
/is unitarily equivalent to the identity representation [7, Lemma 3.2.3] we have that
ϕ
is a
compression of the identity representation. That is, there are isometries
w
:
ℂr→ℂn
and
v
:
ℂs→ℂm
such that
ϕ.x/=w¡xv, ÅxËX.
Since vand ware isometries, we conclude that r≤nand s≤m.
https://doi.org/10.17993/3ctic.2022.112.124-134
(2) Let
ϕ
:
X→n,m
.
ℂ
/be a rectangular matrix extreme state. Then by part .
a
/, there are
isometries (unitaries in this case) uËn.ℂ/,vËm.ℂ/such that
ϕ.x/=v¡xu, ÅxËX.
Conversely let ϕ.x/=v¡xu,ÅxËXfor unitaries uand vthen
S.ϕ/λx
y¡µ=u¡0
0v¡λx
y¡µu0
0v,Åx, y ËX.
Then by [9, Theorem 4.3]
S
.
ϕ
/is pure. Hence
ϕ
is a rectangular matrix extreme state by ( [10,
Proposition2.12]).
The following result is now an immediate consequence of Proposition 2 and Theorem 5.
Theorem 6. Let
Xn,m
.
ℂ
/be an operator space. Then the inclusion map
i
.
x
/=
x,
Å
x
Ë
X
,isa
rectangular matrix extreme state if and only if dim.Xĺ/=1.
REFERENCES
[1]
Aramba
s
ić L.(2005). Irreducible representations of Hilbert C*-modules. Math. Proc. Roy. Irish
Acad., 105 A, 11-14; MR2162903.
[2]
Arunkumar C.S.,Shabna A. M.,Syamkrishnan M. S. and Vijayarajan A. K. (2021).
Extreme states on operator spaces in ternary rings of operators. Proc. Ind. Acad. Sci.131, 44, MR
4338047.
[3] Arveson W. B. (1969). Subalgebras of C¡-algebras. Acta Math. 123, 141-224; MR0253059.
[4]
Arveson W. B.(1972). Subalgebras of
C¡
-algebras
II
.Acta Math. 128(1972) no. 3-4 , 271-308;
MR0394232.
[5]
Arveson W. B.(2011). The noncommutative Choquet boundary II: Hyperrigidity. Israel J. Math.
184 (2011), 349-385; MR2823981.
[6]
Blecher D. P. and Christian Le Merdy (2004). Operator algebras and their modules-an operator
space approach. London Mathematical Society Monographs, New Series, vol. 30, Oxford University
Press, Oxford.
[7]
Bohle D. (2011). K-Theory for ternary structures, Ph.D Thesis, Welstfälishe Wilhelms-Universität
Münster.
[8]
Farenick D. R. (2000). Extremal Matrix states on operator Systems. Journal of London Mathe-
matical Society 61, no. 3, 885-892; MR1766112.
[9]
Farenick D. R.(2004). Pure matrix states on operator systems. Linear Algebra and its Applications
393, 149-173; MR2098611.
[10]
Fuller A. H.,Hartz M. and Lupini M. (2018). Boundary representations of operator spaces,
and compact rectangular matrix convex sets. Journal of Operator Theory, Vol.79, No.1, 139-172;
MR3764146.
[11]
Hamana M. (1999). Triple envelopes and Shilov boundaries of operator spaces. Mathematical
Journal of Toyama University 22, 77-93; MR1744498.
[12]
Kleski C. (2014). Boundary representations and pure completely positive maps. Journal of
Operator Theory, pages 45-62; MR3173052.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
133
[13]
Lobel R. and Paulsen V. I. (1981). Some remarks on
C¡
-convexity. Linear Algebra Appl. 78,
63-78, 1981; MR0599846.
[14]
Magajna B. (2001). On
C¡
-extreme points. Proceedings of the American Mathematical Society
129, 771-780; MR1802000.
[15]
Paulsen V. I. (2002). Completely bounded maps and operator algebras. Cambridge Studies in
Advanced Mathematics Vol. 78, Cambridge University Press, Cambridge.
[16]
Ruan Z. J. (1988). Subspaces of C
¡
-algebras. Journal of Functional Analysis 76, 217-230,
MR0923053.
[17]
Ruan Z. J. (1989). Injectivity and operator spaces. Transactions of the American Mathematical
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[18]
Webster C. and Winkler S. (1999). The Krein-Milman theorem in operator convexity. Tran-
sactions of the American Mathematical Society, Vol.351, no 1, 307-322; MR1615970.
[19]
Wittstock G. (1984). On Matrix Order and Convexity. Functional Analysis: survey and recent
results, III, Math Studies 90, 175-188, MR0761380.
https://doi.org/10.17993/3ctic.2022.112.124-134
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
134