TENSOR PRODUCTS OF WEAK HYPERRIGID SETS
V. A. Anjali
Department of Mathematics, Cochin University of Science And Technology, Ernakulam, Kerala - 682022, India.
E-mail:anjalivnair57@gmail.com
ORCID:
Athul Augustine
Department of Mathematics, Cochin University of Science And Technology, Ernakulam, Kerala - 682022, India.
E-mail:athulaugus@gmail.com
ORCID:
P. Shankar
Department of Mathematics, Cochin University of Science And Technology, Ernakulam, Kerala - 682022, India.
E-mail:shankarsupy@gmail.com, shankarsupy@cusat.ac.in
ORCID:
Reception: 17/10/2022 Acceptance: 01/11/2022 Publication: 29/12/2022
Suggested citation:
V. A. Anjali, Athul Augustine and P. Shankar (2022). Tensor products of weak hyperrigid sets 3C Empresa. Investigación
y pensamiento crítico,11 (2), 164-171. https://doi.org/10.17993/3cemp.2022.110250.164-171
https://doi.org/10.17993/3cemp.2022.110250.164-171
ABSTRACT
In this article, we show that concerning the spatial tensor product of
W∗
-algebras, the tensor product of
two weak hyperrigid operator systems is weak hyperrigid. We prove this result by demonstrating unital
completely positive maps have unique extension property for operator systems if and only if the tensor
product of two unital completely positive maps has unique extension property for the tensor product
of operator systems. Consequently, we prove as a corollary that the tensor product of two boundary
representations for operator systems is boundary representation for the tensor product of operator
systems. The corollary is an analogue result of Hopenwasser’s [9] in the setting of W∗-algebras.
KEYWORDS
Operator system, W∗-algebra, Weak Korovkin set, Boundary representation.
2020 Mathematics Subject Classification: Primary 46L07; Secondary 46L06, 46L89.
https://doi.org/10.17993/3cemp.2022.110250.164-171
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
164
TENSOR PRODUCTS OF WEAK HYPERRIGID SETS
V. A. Anjali
Department of Mathematics, Cochin University of Science And Technology, Ernakulam, Kerala - 682022, India.
E-mail:anjalivnair57@gmail.com
ORCID:
Athul Augustine
Department of Mathematics, Cochin University of Science And Technology, Ernakulam, Kerala - 682022, India.
E-mail:athulaugus@gmail.com
ORCID:
P. Shankar
Department of Mathematics, Cochin University of Science And Technology, Ernakulam, Kerala - 682022, India.
E-mail:shankarsupy@gmail.com, shankarsupy@cusat.ac.in
ORCID:
Reception: 17/10/2022 Acceptance: 01/11/2022 Publication: 29/12/2022
Suggested citation:
V. A. Anjali, Athul Augustine and P. Shankar (2022). Tensor products of weak hyperrigid sets 3C Empresa. Investigación
y pensamiento crítico,11 (2), 164-171. https://doi.org/10.17993/3cemp.2022.110250.164-171
https://doi.org/10.17993/3cemp.2022.110250.164-171
ABSTRACT
In this article, we show that concerning the spatial tensor product of
W∗
-algebras, the tensor product of
two weak hyperrigid operator systems is weak hyperrigid. We prove this result by demonstrating unital
completely positive maps have unique extension property for operator systems if and only if the tensor
product of two unital completely positive maps has unique extension property for the tensor product
of operator systems. Consequently, we prove as a corollary that the tensor product of two boundary
representations for operator systems is boundary representation for the tensor product of operator
systems. The corollary is an analogue result of Hopenwasser’s [9] in the setting of W∗-algebras.
KEYWORDS
Operator system, W∗-algebra, Weak Korovkin set, Boundary representation.
2020 Mathematics Subject Classification: Primary 46L07; Secondary 46L06, 46L89.
https://doi.org/10.17993/3cemp.2022.110250.164-171
165
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
1 INTRODUCTION
Positive approximation processes play a fundamental role in approximation theory and appear naturally
in many problems. In 1953, Korovkin [11] discovered the most powerful and, at the same time, the
simplest criterion to decide whether a given sequence
{ϕn}n∈N
of positive linear operators on the
space of complex-valued continuous functions
C
(
X
), where
X
is a compact Hausdorff space is an
approximation process. That is,
ϕn
(
f
)
→f
uniformly on
X
for every
f∈C
(
X
). In fact it is sufficient to
verify that
ϕn
(
f
)
→f
uniformly on
X
only for
f∈{
1
, x, x2}
. This set is called a Korovkin set. Starting
with this result, during the last thirty years, many mathematicians have extended Korovkin’s theorem
to other function spaces or, more generally, to abstract spaces such as Banach algebras, Banach spaces,
C∗
-algebras and so on. At the same time, strong and fruitful connections of this theory have been
revealed with classical approximation theory and other fields such as Choquet boundaries, convexity
theory, uniqueness of extensions of positive linear maps, and so on.
Here, we provide an expository review of the non-commutative analogue of Korovkin’s theorems with
weak operator convergence and norm convergence. The notion of boundary representation of a
C∗
-algebra
for an operator system introduced by Arveson [2] greatly influenced the theory of noncommutative
approximation theory and other related areas such as Korovkin type properties for completely positive
maps, peaking phenomena for operator systems and noncommutative convexity, etc. Arveson [4]
introduced the notion of hyperrigid set as a noncommutative analogue of classical Korovkin set and
studied the relation between hyperrigid operator systems and boundary representations extensively.
In 1984, Limaye and Namboodiri [13] studied the non-commutative Korovkin sets on
B
(
H
)using
weak operator convergence, which they named weak Korovkin sets. Limaye and Namboodiri [13] proved
an exciting result using a famous boundary theorem of Arveson [3] that is as follows: An irreducible
subset of
B
(
H
)containing identity and a nonzero compact operator is weak Korovkin in
B
(
H
)if and
only if the identity representation of the
C∗
-algebra generated by the irreducible set has a unique
completely positive linear extension to the
C∗
-algebra when restricted to the irreducible set. Limaye
and Namboodiri gave many examples to establish these notions and theorems.
Namboodiri, inspired by Arveson’s paper [4] on hyperrigidity, modified [15] the notion of weak
Korovkin set on
B
(
H
)to weak hyperrigid set in the context of
W∗
-algebras using weak operator
convergence. He generalized the theorem in [13], characterizing the weak Korovkin set without assuming
the presence of compact operators, and explored all nondegenerate representations. The result is as
follows: An operator system is weak hyperrigid in the
W∗
-algebra generated by it if and only if every
nondegenerate representation has a unique completely positive linear extension to the
W∗
-algebra
when restricted to the operator system. Using this theorem, he established the partial answer to the
non-commutative analogue of Saskin’s theorem [18] relating weak hyperrigidity and Choquet boundary.
Namboodiri gave a brief survey of the developments in the ‘non-commutative Korovkin-type theory’
in [14]. Namboodiri, Pramod, Shankar, and Vijayarajan [16] studied the non-commutative analogue of
Saskin’s theorem using the notions quasi hyperrigidity and weak boundary representations. Shankar
and Vijayarajan [21] proved that the tensor product of two hyperrigid operator systems is hyperrigid in
the spatial tensor product of
C∗
-algebras. Arunkumar and Vijayarajan [1] studied the tensor products
of quasi hyperrigid operator systems introduced in [16]. Shankar [19] established hyperrigid generators
for certain C∗-algebras.
In this article, we study the weak hyperrigidity of operator systems in
W∗
-algebras in the context
of tensor products of
W∗
-algebras. It is interesting to investigate whether the tensor product of weak
hyperrigid operator systems is weak hyperrigid. As a result of Hopenwasser [9], the tensor product of
boundary representations of
C∗
-algebras for operator systems is a boundary representation if one of
the constituent
C∗
-algebras is a GCR algebra. Since weak hyperrigidity implies that all irreducible
representations are boundary representations for
W∗
-algebra, we will be able to deduce Hopenwasser’s
result for
W∗
-algebras as a spacial case. We achieve this by establishing first that unique extension
property for unital completely positive maps on operator systems carry over to the tensor product
of those maps defined on the tensor product of operator systems in the spatial tensor product of
W∗-algebras.
https://doi.org/10.17993/3cemp.2022.110250.164-171
2 PRELIMINARIES
To fix our notation and terminology, we recall the fundamental notions. Let
H
be a complex Hilbert
space and let
B
(
H
)be the bounded linear operators on
H
. An operator system
S
in a
W∗
-algebra
M
is a self-adjoint linear subspace of
M
containing the identity of
M
. An operator algebra
A
in a
W∗-algebra Mis a subalgebra of Mcontaining the identity of M.
Let
ϕ
be a linear map from a
W∗
-algebra
M
into a
W∗
-algebra
N
, we can define a family of maps
ϕn
:
Mn
(
M
)
→Mn
(
N
)given by
ϕn
([
aij
]) = [
ϕ
(
aij
)]. We say that
ϕ
is completely bounded (CB) if
||ϕ||CB = supn≥1||ϕn|| <∞. We say that ϕis completely contractive (CC) if ||ϕ||CB ≤1and that ϕis
completely isometric if
ϕn
is isometric for all
n≥
1. We say that
ϕ
is completely positive (CP) if
ϕn
is
positive for all n≥1, and that ϕis unital completely positive (UCP) if in addition ϕ(1)=1.
Definition 1. [2] Let
S
be an operator system in a
W∗
-algebra
M
. A nondegenerate representation
π
:
M→B
(
H
)has a unique extension property (UEP) for
S
if
π|S
has a unique completely positive
extension, namely
π
itself to
M
. If
π
is an irreducible representation, then
π
is said to be a boundary
representation for S.
Definition 2. [15] A set
G
of generators of a
W∗
-algebra
M
containing the identity 1
M
is said to be
weak hyperrigid if for every faithful representation
M⊆B
(
H
)of
M
on a separable Hilbert space
H
and every net {ϕα}α∈Iof contractive completely positive maps from B(H)to itself.
lim
αϕα(g)=gweakly ∀g∈G=⇒lim
αϕα(a)=aweakly ∀a∈M.
Theorem 1. [15] For every separable operator system
S
, that generates a
W∗
-algebra
M
, the following
are equivalent.
(i) Sis weak hyperrigid.
(ii)
For every nondegenerate representation
π
:
M→B
(
H
), on a separable Hilbert space
H
and every
net {ϕα}α∈Iof contractive completely positive maps from Mto B(H).
limαϕα(s)=π(s)weakly ∀s∈S=⇒limαϕα(a)=π(a)weakly ∀a∈M.
(iii)
For every nondegenerate representation
π
:
M→B
(
H
)on a separable Hilbert space
H
,
π|S
has a
unique extension property.
(iv)
For every
W∗
-algebra
N
, every homomorphism
θ
:
M→N
such that
θ
(1
M
)=1
N
and every
contractive completely positive map ϕ:N→N,
ϕ(x)=x∀x∈θ(S)=⇒ϕ(x)=x∀x∈θ(M).
In this context, mentioning the ‘hyperrigidity conjecture’ posed by Arveson [4] is relevant. The
hyperrigidity conjecture states that if every irreducible representation of a
C∗
-algebra
A
is a boundary
representation for a separable operator system
S⊆A
and
A
=
C∗
(
S
), then
S
is hyperrigid. Arveson [4]
proved the conjecture for
C∗
-algebras having a countable spectrum, while Kleski [10] established
the conjecture for all type-I
C∗
-algebras with some additional assumptions. Recently Davidson and
Kennedy [6] proved the conjecture for function systems.
Using the apparent correspondence between representations and modules, one can translate many
aspects of the above notions into Hilbert modules. Muhly and Solel [12] gave an algebraic characterization
of boundary representations in terms of Hilbert modules. Following Muhly and Solel, Shankar and
Vijayarajan [20,22] established a Hilbert module characterization for hyperrigidity (weak hyperrigidity)
of specific operator systems in a C∗-algebra (W∗-algebras).
We need to consider tensor products of
W∗
-algebras in this article. Let
A1⊗A2
denote the algebraic
tensor product of
A1
and
A2
. Let
A1⊗sA2
denote the closure of
A1⊗A2
provided with the spatial
norm, which is the minimal
C∗
-norm on the tensor product of
W∗
-algebras. In what follows, we will
consider the spatial norm for the tensor product of
W∗
-algebras. We know that if representations
π1
is
nondegenerate on
A1
and
π2
is nondegenerate on
A2
, then the representation
π1⊗π2
is nondegenerate
https://doi.org/10.17993/3cemp.2022.110250.164-171
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
166
1 INTRODUCTION
Positive approximation processes play a fundamental role in approximation theory and appear naturally
in many problems. In 1953, Korovkin [11] discovered the most powerful and, at the same time, the
simplest criterion to decide whether a given sequence
{ϕn}n∈N
of positive linear operators on the
space of complex-valued continuous functions
C
(
X
), where
X
is a compact Hausdorff space is an
approximation process. That is,
ϕn
(
f
)
→f
uniformly on
X
for every
f∈C
(
X
). In fact it is sufficient to
verify that
ϕn
(
f
)
→f
uniformly on
X
only for
f∈{
1
, x, x2}
. This set is called a Korovkin set. Starting
with this result, during the last thirty years, many mathematicians have extended Korovkin’s theorem
to other function spaces or, more generally, to abstract spaces such as Banach algebras, Banach spaces,
C∗
-algebras and so on. At the same time, strong and fruitful connections of this theory have been
revealed with classical approximation theory and other fields such as Choquet boundaries, convexity
theory, uniqueness of extensions of positive linear maps, and so on.
Here, we provide an expository review of the non-commutative analogue of Korovkin’s theorems with
weak operator convergence and norm convergence. The notion of boundary representation of a
C∗
-algebra
for an operator system introduced by Arveson [2] greatly influenced the theory of noncommutative
approximation theory and other related areas such as Korovkin type properties for completely positive
maps, peaking phenomena for operator systems and noncommutative convexity, etc. Arveson [4]
introduced the notion of hyperrigid set as a noncommutative analogue of classical Korovkin set and
studied the relation between hyperrigid operator systems and boundary representations extensively.
In 1984, Limaye and Namboodiri [13] studied the non-commutative Korovkin sets on
B
(
H
)using
weak operator convergence, which they named weak Korovkin sets. Limaye and Namboodiri [13] proved
an exciting result using a famous boundary theorem of Arveson [3] that is as follows: An irreducible
subset of
B
(
H
)containing identity and a nonzero compact operator is weak Korovkin in
B
(
H
)if and
only if the identity representation of the
C∗
-algebra generated by the irreducible set has a unique
completely positive linear extension to the
C∗
-algebra when restricted to the irreducible set. Limaye
and Namboodiri gave many examples to establish these notions and theorems.
Namboodiri, inspired by Arveson’s paper [4] on hyperrigidity, modified [15] the notion of weak
Korovkin set on
B
(
H
)to weak hyperrigid set in the context of
W∗
-algebras using weak operator
convergence. He generalized the theorem in [13], characterizing the weak Korovkin set without assuming
the presence of compact operators, and explored all nondegenerate representations. The result is as
follows: An operator system is weak hyperrigid in the
W∗
-algebra generated by it if and only if every
nondegenerate representation has a unique completely positive linear extension to the
W∗
-algebra
when restricted to the operator system. Using this theorem, he established the partial answer to the
non-commutative analogue of Saskin’s theorem [18] relating weak hyperrigidity and Choquet boundary.
Namboodiri gave a brief survey of the developments in the ‘non-commutative Korovkin-type theory’
in [14]. Namboodiri, Pramod, Shankar, and Vijayarajan [16] studied the non-commutative analogue of
Saskin’s theorem using the notions quasi hyperrigidity and weak boundary representations. Shankar
and Vijayarajan [21] proved that the tensor product of two hyperrigid operator systems is hyperrigid in
the spatial tensor product of
C∗
-algebras. Arunkumar and Vijayarajan [1] studied the tensor products
of quasi hyperrigid operator systems introduced in [16]. Shankar [19] established hyperrigid generators
for certain C∗-algebras.
In this article, we study the weak hyperrigidity of operator systems in
W∗
-algebras in the context
of tensor products of
W∗
-algebras. It is interesting to investigate whether the tensor product of weak
hyperrigid operator systems is weak hyperrigid. As a result of Hopenwasser [9], the tensor product of
boundary representations of
C∗
-algebras for operator systems is a boundary representation if one of
the constituent
C∗
-algebras is a GCR algebra. Since weak hyperrigidity implies that all irreducible
representations are boundary representations for
W∗
-algebra, we will be able to deduce Hopenwasser’s
result for
W∗
-algebras as a spacial case. We achieve this by establishing first that unique extension
property for unital completely positive maps on operator systems carry over to the tensor product
of those maps defined on the tensor product of operator systems in the spatial tensor product of
W∗-algebras.
https://doi.org/10.17993/3cemp.2022.110250.164-171
2 PRELIMINARIES
To fix our notation and terminology, we recall the fundamental notions. Let
H
be a complex Hilbert
space and let
B
(
H
)be the bounded linear operators on
H
. An operator system
S
in a
W∗
-algebra
M
is a self-adjoint linear subspace of
M
containing the identity of
M
. An operator algebra
A
in a
W∗-algebra Mis a subalgebra of Mcontaining the identity of M.
Let
ϕ
be a linear map from a
W∗
-algebra
M
into a
W∗
-algebra
N
, we can define a family of maps
ϕn
:
Mn
(
M
)
→Mn
(
N
)given by
ϕn
([
aij
]) = [
ϕ
(
aij
)]. We say that
ϕ
is completely bounded (CB) if
||ϕ||CB = supn≥1||ϕn|| <∞. We say that ϕis completely contractive (CC) if ||ϕ||CB ≤1and that ϕis
completely isometric if
ϕn
is isometric for all
n≥
1. We say that
ϕ
is completely positive (CP) if
ϕn
is
positive for all n≥1, and that ϕis unital completely positive (UCP) if in addition ϕ(1)=1.
Definition 1. [2] Let
S
be an operator system in a
W∗
-algebra
M
. A nondegenerate representation
π
:
M→B
(
H
)has a unique extension property (UEP) for
S
if
π|S
has a unique completely positive
extension, namely
π
itself to
M
. If
π
is an irreducible representation, then
π
is said to be a boundary
representation for S.
Definition 2. [15] A set
G
of generators of a
W∗
-algebra
M
containing the identity 1
M
is said to be
weak hyperrigid if for every faithful representation
M⊆B
(
H
)of
M
on a separable Hilbert space
H
and every net {ϕα}α∈Iof contractive completely positive maps from B(H)to itself.
lim
αϕα(g)=gweakly ∀g∈G=⇒lim
αϕα(a)=aweakly ∀a∈M.
Theorem 1. [15] For every separable operator system
S
, that generates a
W∗
-algebra
M
, the following
are equivalent.
(i) Sis weak hyperrigid.
(ii)
For every nondegenerate representation
π
:
M→B
(
H
), on a separable Hilbert space
H
and every
net {ϕα}α∈Iof contractive completely positive maps from Mto B(H).
limαϕα(s)=π(s)weakly ∀s∈S=⇒limαϕα(a)=π(a)weakly ∀a∈M.
(iii)
For every nondegenerate representation
π
:
M→B
(
H
)on a separable Hilbert space
H
,
π|S
has a
unique extension property.
(iv)
For every
W∗
-algebra
N
, every homomorphism
θ
:
M→N
such that
θ
(1
M
)=1
N
and every
contractive completely positive map ϕ:N→N,
ϕ(x)=x∀x∈θ(S)=⇒ϕ(x)=x∀x∈θ(M).
In this context, mentioning the ‘hyperrigidity conjecture’ posed by Arveson [4] is relevant. The
hyperrigidity conjecture states that if every irreducible representation of a
C∗
-algebra
A
is a boundary
representation for a separable operator system
S⊆A
and
A
=
C∗
(
S
), then
S
is hyperrigid. Arveson [4]
proved the conjecture for
C∗
-algebras having a countable spectrum, while Kleski [10] established
the conjecture for all type-I
C∗
-algebras with some additional assumptions. Recently Davidson and
Kennedy [6] proved the conjecture for function systems.
Using the apparent correspondence between representations and modules, one can translate many
aspects of the above notions into Hilbert modules. Muhly and Solel [12] gave an algebraic characterization
of boundary representations in terms of Hilbert modules. Following Muhly and Solel, Shankar and
Vijayarajan [20,22] established a Hilbert module characterization for hyperrigidity (weak hyperrigidity)
of specific operator systems in a C∗-algebra (W∗-algebras).
We need to consider tensor products of
W∗
-algebras in this article. Let
A1⊗A2
denote the algebraic
tensor product of
A1
and
A2
. Let
A1⊗sA2
denote the closure of
A1⊗A2
provided with the spatial
norm, which is the minimal
C∗
-norm on the tensor product of
W∗
-algebras. In what follows, we will
consider the spatial norm for the tensor product of
W∗
-algebras. We know that if representations
π1
is
nondegenerate on
A1
and
π2
is nondegenerate on
A2
, then the representation
π1⊗π2
is nondegenerate
https://doi.org/10.17993/3cemp.2022.110250.164-171
167
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
on
A1⊗A2
. Conversely, from [5, Theorem II.9.2.1] and [17, Proposition 1.22.11] we can see that if
π
is
a nondegenerate representation of
A1⊗A2
, then there are unique nondegenerate representations
π1
of
A1and π2of A2such that π=π1⊗π2.
Tensor products of operator spaces (linear subspaces) of
C∗
-algebras and operator spaces of tensor
product of
C∗
-algebras were explored by Hopenwasser earlier in [8], and [9] to study boundary
representations. In [8], it was shown that boundary representations of an operator subspace of a
C∗
-algebra
A⊗Mn
(
C
)under certain conditions are parameterized by the boundary representations
of an operator subspace of the
C∗
-algebra
A
which is given by the operator subspace in
A⊗Mn
(
C
).
In [9], it was proved that if one of the
C∗
-algebras of the tensor product is a GCR algebra, then the
boundary representations of the tensor product of
C∗
-algebras correspond to products of boundary
representations.
3 MAIN RESULTS
In our main result, we investigate the relationship between the weak hyperrigidity of the tensor product
of two operator systems in the tensor product
W∗
-algebra and the weak hyperrigidity of the individual
operator systems in the respective
W∗
-algebras. The following result shows that the unique extension
property of completely positive maps on operator systems carries over to the tensor product of those
maps defined on the tensor product of operator systems.
Theorem 2. Let
S1
and
S2
be operator systems generating
W∗
-algebras
A1
and
A2
respectively. Let
πi
:
Si→B
(
Hi
)
,i
=1
,
2be unital completely positive maps. Then
π1
and
π2
have unique extension
property if and only if the unital completely positive map
π1⊗π2
:
S1⊗S2→B
(
H1⊗H2
)has unique
extension property for S1⊗S2⊆A1⊗sA2.
Proof. Assume that
π1⊗π2
has unique extension property, that is
π1⊗π2
has unique completely
positive extension
π1⊗sπ2
:
A1⊗sA2→B
(
H1⊗H2
)which is a representation of
A1⊗sA2
. We will
show that
π1
and
π2
have unique extension property. On the contrary, assume that one of the factors,
say
π1
does not have unique extension property. This means that there exist at least two extensions
of
π1
, a completely positive map
ϕ1
:
A1→B
(
H1
)and the representation
π1
:
A1→B
(
H1
)such
that
ϕ1
=
π1
on
A1
, but
ϕ1
=
π1
=
π1
on
S1
. Using [5, II.9.7], we can see that the tensor product
of two completely positive maps is completely positive. We have
ϕ1⊗sπ2
is a completely positive
extension of
π1⊗π2
on
S1⊗S2
, where
π2
is a unique completely positive extension of
π2
on
S2
. Hence
ϕ1⊗sπ2=π1⊗sπ2on A1⊗sA2. This contradicts our assumption.
Conversely, assume that
π1
and
π2
have the unique extension property, that is
π1
and
π2
have unique
completely positive extensions
π1
:
A1→B
(
H1
)and
π2
:
A2→B
(
H2
)respectively where
π1
and
π2
are representations of
A1
and
A2
respectively. We will show that
π1⊗π2
has the unique extension
property. We have
π1⊗sπ2
:
A1⊗sA2→B
(
H1⊗H2
)is a representation and an extension of
π1⊗π2
on
S1⊗S2
. It is enough to show that if
ϕ
:
A1⊗sA2→B
(
H1⊗H2
)is a completely positive extension
of π1⊗π2on S1⊗S2then ϕ=π1⊗sπ2on A1⊗A2.
Let
P
be any rank one projection in
B
(
H2
). The map
a→
(1
⊗P
)
ϕ
(
a⊗
1)(1
⊗P
)is completely positive
on
A1
, since the map is a composition of three completely positive maps. Let
v
be a unit vector in the
range of
P
and let
K
be the range of 1
⊗P
. Define
U
:
H1→K
by
U
(
x
)=
x⊗v, x ∈H1
,
U
is a unitary
map. Let
ˆπ
=
Uπ1
(
a
)
U∗,a∈A1
and
ˆπ
(
a
)is the restriction to
K
of
π1
(
a
)
⊗P
= (1
⊗P
)(
π1
(
a
)
⊗
1)(1
⊗P
).
Since
ˆπ
is unitarily equivalent to
π1
, the representation
ˆπ|S1
has unique extension property. Let
ψ
(
a
)
be the restriction to
K
of (1
⊗P
)
ϕ
(
a⊗
1)(1
⊗P
)which implies that
ψ
is a completely positive map
that agrees with ˆπon S1, hence on all of A1.
Let
x, y ∈H1
and
r∈H2
. From the above paragraph we have, for any
a∈A1
,
⟨ϕ(a⊗1)(x⊗r),y⊗r⟩
=
⟨(π1(a)⊗1)(x⊗r),y⊗r⟩
. (Letting
P
be the rank one projection on the
subspace spanned by
r
.) Let
D
=
ϕ
(
a⊗
1)
−π1⊗
1. Then we have
⟨D(x⊗r),y⊗r⟩
=0, for all
x, y ∈H1,r∈H2. Using polarization formula
https://doi.org/10.17993/3cemp.2022.110250.164-171
4⟨D(x⊗r),y⊗s⟩=⟨D(x⊗(r+s)),y⊗(r+s)⟩
−⟨D(x⊗(r−s)),y⊗(r−s)⟩
+i⟨D(x⊗(r+is)),y⊗(r+is)⟩
−i⟨D(x⊗(r−is)),y⊗(r−is)⟩.
We have
⟨D(x⊗r),y⊗s⟩
=0, for all
x, y ∈H1
and for all
r, s ∈H2
. Consequently, if
z1
=
n
i=1
xi⊗ri
and
z2
=
m
i=1
yi⊗si
, then
⟨Dz1,z
2⟩
=0. Since
z1,z
2
run through a dense subset of
H1⊗H2
and
D
is bounded,
D
=0. Therefore
ϕ
(
a⊗
1) =
π1
(
a
)
⊗
1, for all
a∈A1
. In the same way we can obtain
ϕ
(1
⊗b
)=1
⊗π2
(
b
), for all
b∈A2
. Since
ϕ
is a completely positive map on
A1⊗A2
and
ϕ
(1
⊗b
)=1
⊗π2
(
b
),
for all b∈A2, using a multiplicative domain argument, e.g., see [9, Lemma 2] we have
ϕ(a⊗b)=ϕ(a⊗1)(1 ⊗π2(b)) = (1 ⊗π2(b))ϕ(a⊗1)
for all a∈A1,b∈A2. Also ϕ(a⊗1) = π1(a)⊗1, for all a∈A1. Hence ϕ=π1⊗sπ2on A1⊗sA2.
Corollary 1. Let
S1
and
S2
be separable operator systems generating
W∗
-algebras
A1
and
A2
respectively.
Assume that either
A1
or
A2
is a GCR algebra. Then
S1
and
S2
are weak hyperrigid in
A1
and
A2
respectively if and only if S1⊗S2is weak hyperrigid in A1⊗sA2.
Proof. Assume that
S1⊗S2
is weak hyperrigid in the
W∗
-algebra
A1⊗sA2
. By theorem 1, every
unital representation
π
:
A1⊗sA2→B
(
H1⊗H2
),
π|S1⊗S2
has unique extension property. We have if
π
is a unital representation of
A1⊗sA2
, since one of the
W∗
-algebras is GCR then by [7, Proposition 2]
there are unique unital representations
π1
of
A1
and
π2
of
A2
such that
π
=
π1⊗sπ2
. Using theorem 2,
we can see that
π1|S1
and
π2|S2
have unique extension property. This implies that
S1
and
S2
are weak
hyperrigid in A1and A2respectively again by theorem 1.
Conversely, assume that
S1
is weak hyperrigid in
A1
and
S2
is weak hyperrigid in
A2
. By theorem 1,
for every unital representations
π1
:
A1→B
(
H1
)and
π2
:
A2→B
(
H2
),
π1|S1
and
π2|S2
have unique
extension property. We have, if
π1
and
π2
are unital representations of
A1
and
A2
respectively, then
π1⊗sπ2is an unital representation of A1⊗sA2. Using theorem 2, we can see that π1⊗sπ2|S1⊗S2has
unique extension property. Now, by theorem 1 S1⊗S2is weak hyperrigid in A1⊗sA2.
Let
A1⊗mA2
denote the closure of
A1⊗A2
provided with maximal
C∗
-norm. There are
C∗
-algebras
A1
for which the minimal and the maximal norm on
A1⊗A2
coincide for all
C∗
-algebras
A2
and
consequently the
C∗
-norm on
A1⊗A2
is unique. Such
C∗
-algebras are called nuclear. The spatial norm
assumption in the above results is redundant if the
C∗
-algebras are nuclear. But general
C∗
-algebras
with the lack of injectivity associated with other
C∗
-norms, including the maximal one, will require
additional assumptions.
Let
A1
and
A2
be
W∗
-algebras and
γ
is any
C∗
-cross norm on
A1⊗A2
. If
π1
and
π2
are irreducible
representations of
A1
and
A2
respectively, then
π1⊗γπ2
is an irreducible representation of
A1⊗γA2
.
Conversely, every irreducible representation
π
on
A1⊗γA2
need not factor as a product
π1⊗γπ2
of irreducible representations. If we assume, one of the
W∗
-algebra is a GCR algebra, and then
by [7, Proposition 2], every irreducible representation does factor. Since GCR algebras are nuclear,
there is a unique C∗-cross norm on A1⊗A2, which we denote by A1⊗γA2.
Using the above facts, the result by Hopenwasser [9] relating boundary representations of tensor
products of C∗-algebras will become a corollary to our theorem 2.
Corollary 2. Let
S1
and
S2
be unital operator subspaces of generating
W∗
-algebras
A1
and
A2
respectively. Assume that either
A1
or
A2
is a GCR algebra. Then the representation
π1⊗γπ2
of
A1⊗γA2
is a boundary representation for
S1⊗S2
if and only if the representations
π1
of
A1
and
π2
of
A2are boundary representations for S1and S2respectively.
Now, we will provide some examples which illustrate the results above.
https://doi.org/10.17993/3cemp.2022.110250.164-171
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
168
on
A1⊗A2
. Conversely, from [5, Theorem II.9.2.1] and [17, Proposition 1.22.11] we can see that if
π
is
a nondegenerate representation of
A1⊗A2
, then there are unique nondegenerate representations
π1
of
A1and π2of A2such that π=π1⊗π2.
Tensor products of operator spaces (linear subspaces) of
C∗
-algebras and operator spaces of tensor
product of
C∗
-algebras were explored by Hopenwasser earlier in [8], and [9] to study boundary
representations. In [8], it was shown that boundary representations of an operator subspace of a
C∗
-algebra
A⊗Mn
(
C
)under certain conditions are parameterized by the boundary representations
of an operator subspace of the
C∗
-algebra
A
which is given by the operator subspace in
A⊗Mn
(
C
).
In [9], it was proved that if one of the
C∗
-algebras of the tensor product is a GCR algebra, then the
boundary representations of the tensor product of
C∗
-algebras correspond to products of boundary
representations.
3 MAIN RESULTS
In our main result, we investigate the relationship between the weak hyperrigidity of the tensor product
of two operator systems in the tensor product
W∗
-algebra and the weak hyperrigidity of the individual
operator systems in the respective
W∗
-algebras. The following result shows that the unique extension
property of completely positive maps on operator systems carries over to the tensor product of those
maps defined on the tensor product of operator systems.
Theorem 2. Let
S1
and
S2
be operator systems generating
W∗
-algebras
A1
and
A2
respectively. Let
πi
:
Si→B
(
Hi
)
,i
=1
,
2be unital completely positive maps. Then
π1
and
π2
have unique extension
property if and only if the unital completely positive map
π1⊗π2
:
S1⊗S2→B
(
H1⊗H2
)has unique
extension property for S1⊗S2⊆A1⊗sA2.
Proof. Assume that
π1⊗π2
has unique extension property, that is
π1⊗π2
has unique completely
positive extension
π1⊗sπ2
:
A1⊗sA2→B
(
H1⊗H2
)which is a representation of
A1⊗sA2
. We will
show that
π1
and
π2
have unique extension property. On the contrary, assume that one of the factors,
say
π1
does not have unique extension property. This means that there exist at least two extensions
of
π1
, a completely positive map
ϕ1
:
A1→B
(
H1
)and the representation
π1
:
A1→B
(
H1
)such
that
ϕ1
=
π1
on
A1
, but
ϕ1
=
π1
=
π1
on
S1
. Using [5, II.9.7], we can see that the tensor product
of two completely positive maps is completely positive. We have
ϕ1⊗sπ2
is a completely positive
extension of
π1⊗π2
on
S1⊗S2
, where
π2
is a unique completely positive extension of
π2
on
S2
. Hence
ϕ1⊗sπ2=π1⊗sπ2on A1⊗sA2. This contradicts our assumption.
Conversely, assume that
π1
and
π2
have the unique extension property, that is
π1
and
π2
have unique
completely positive extensions
π1
:
A1→B
(
H1
)and
π2
:
A2→B
(
H2
)respectively where
π1
and
π2
are representations of
A1
and
A2
respectively. We will show that
π1⊗π2
has the unique extension
property. We have
π1⊗sπ2
:
A1⊗sA2→B
(
H1⊗H2
)is a representation and an extension of
π1⊗π2
on
S1⊗S2
. It is enough to show that if
ϕ
:
A1⊗sA2→B
(
H1⊗H2
)is a completely positive extension
of π1⊗π2on S1⊗S2then ϕ=π1⊗sπ2on A1⊗A2.
Let
P
be any rank one projection in
B
(
H2
). The map
a→
(1
⊗P
)
ϕ
(
a⊗
1)(1
⊗P
)is completely positive
on
A1
, since the map is a composition of three completely positive maps. Let
v
be a unit vector in the
range of
P
and let
K
be the range of 1
⊗P
. Define
U
:
H1→K
by
U
(
x
)=
x⊗v, x ∈H1
,
U
is a unitary
map. Let
ˆπ
=
Uπ1
(
a
)
U∗,a∈A1
and
ˆπ
(
a
)is the restriction to
K
of
π1
(
a
)
⊗P
= (1
⊗P
)(
π1
(
a
)
⊗
1)(1
⊗P
).
Since
ˆπ
is unitarily equivalent to
π1
, the representation
ˆπ|S1
has unique extension property. Let
ψ
(
a
)
be the restriction to
K
of (1
⊗P
)
ϕ
(
a⊗
1)(1
⊗P
)which implies that
ψ
is a completely positive map
that agrees with ˆπon S1, hence on all of A1.
Let
x, y ∈H1
and
r∈H2
. From the above paragraph we have, for any
a∈A1
,
⟨ϕ(a⊗1)(x⊗r),y⊗r⟩
=
⟨(π1(a)⊗1)(x⊗r),y⊗r⟩
. (Letting
P
be the rank one projection on the
subspace spanned by
r
.) Let
D
=
ϕ
(
a⊗
1)
−π1⊗
1. Then we have
⟨D(x⊗r),y⊗r⟩
=0, for all
x, y ∈H1,r∈H2. Using polarization formula
https://doi.org/10.17993/3cemp.2022.110250.164-171
4⟨D(x⊗r),y⊗s⟩=⟨D(x⊗(r+s)),y⊗(r+s)⟩
−⟨D(x⊗(r−s)),y⊗(r−s)⟩
+i⟨D(x⊗(r+is)),y⊗(r+is)⟩
−i⟨D(x⊗(r−is)),y⊗(r−is)⟩.
We have
⟨D(x⊗r),y⊗s⟩
=0, for all
x, y ∈H1
and for all
r, s ∈H2
. Consequently, if
z1
=
n
i=1
xi⊗ri
and
z2
=
m
i=1
yi⊗si
, then
⟨Dz1,z
2⟩
=0. Since
z1,z
2
run through a dense subset of
H1⊗H2
and
D
is bounded,
D
=0. Therefore
ϕ
(
a⊗
1) =
π1
(
a
)
⊗
1, for all
a∈A1
. In the same way we can obtain
ϕ
(1
⊗b
)=1
⊗π2
(
b
), for all
b∈A2
. Since
ϕ
is a completely positive map on
A1⊗A2
and
ϕ
(1
⊗b
)=1
⊗π2
(
b
),
for all b∈A2, using a multiplicative domain argument, e.g., see [9, Lemma 2] we have
ϕ(a⊗b)=ϕ(a⊗1)(1 ⊗π2(b)) = (1 ⊗π2(b))ϕ(a⊗1)
for all a∈A1,b∈A2. Also ϕ(a⊗1) = π1(a)⊗1, for all a∈A1. Hence ϕ=π1⊗sπ2on A1⊗sA2.
Corollary 1. Let
S1
and
S2
be separable operator systems generating
W∗
-algebras
A1
and
A2
respectively.
Assume that either
A1
or
A2
is a GCR algebra. Then
S1
and
S2
are weak hyperrigid in
A1
and
A2
respectively if and only if S1⊗S2is weak hyperrigid in A1⊗sA2.
Proof. Assume that
S1⊗S2
is weak hyperrigid in the
W∗
-algebra
A1⊗sA2
. By theorem 1, every
unital representation
π
:
A1⊗sA2→B
(
H1⊗H2
),
π|S1⊗S2
has unique extension property. We have if
π
is a unital representation of
A1⊗sA2
, since one of the
W∗
-algebras is GCR then by [7, Proposition 2]
there are unique unital representations
π1
of
A1
and
π2
of
A2
such that
π
=
π1⊗sπ2
. Using theorem 2,
we can see that
π1|S1
and
π2|S2
have unique extension property. This implies that
S1
and
S2
are weak
hyperrigid in A1and A2respectively again by theorem 1.
Conversely, assume that
S1
is weak hyperrigid in
A1
and
S2
is weak hyperrigid in
A2
. By theorem 1,
for every unital representations
π1
:
A1→B
(
H1
)and
π2
:
A2→B
(
H2
),
π1|S1
and
π2|S2
have unique
extension property. We have, if
π1
and
π2
are unital representations of
A1
and
A2
respectively, then
π1⊗sπ2is an unital representation of A1⊗sA2. Using theorem 2, we can see that π1⊗sπ2|S1⊗S2has
unique extension property. Now, by theorem 1 S1⊗S2is weak hyperrigid in A1⊗sA2.
Let
A1⊗mA2
denote the closure of
A1⊗A2
provided with maximal
C∗
-norm. There are
C∗
-algebras
A1
for which the minimal and the maximal norm on
A1⊗A2
coincide for all
C∗
-algebras
A2
and
consequently the
C∗
-norm on
A1⊗A2
is unique. Such
C∗
-algebras are called nuclear. The spatial norm
assumption in the above results is redundant if the
C∗
-algebras are nuclear. But general
C∗
-algebras
with the lack of injectivity associated with other
C∗
-norms, including the maximal one, will require
additional assumptions.
Let
A1
and
A2
be
W∗
-algebras and
γ
is any
C∗
-cross norm on
A1⊗A2
. If
π1
and
π2
are irreducible
representations of
A1
and
A2
respectively, then
π1⊗γπ2
is an irreducible representation of
A1⊗γA2
.
Conversely, every irreducible representation
π
on
A1⊗γA2
need not factor as a product
π1⊗γπ2
of irreducible representations. If we assume, one of the
W∗
-algebra is a GCR algebra, and then
by [7, Proposition 2], every irreducible representation does factor. Since GCR algebras are nuclear,
there is a unique C∗-cross norm on A1⊗A2, which we denote by A1⊗γA2.
Using the above facts, the result by Hopenwasser [9] relating boundary representations of tensor
products of C∗-algebras will become a corollary to our theorem 2.
Corollary 2. Let
S1
and
S2
be unital operator subspaces of generating
W∗
-algebras
A1
and
A2
respectively. Assume that either
A1
or
A2
is a GCR algebra. Then the representation
π1⊗γπ2
of
A1⊗γA2
is a boundary representation for
S1⊗S2
if and only if the representations
π1
of
A1
and
π2
of
A2are boundary representations for S1and S2respectively.
Now, we will provide some examples which illustrate the results above.
https://doi.org/10.17993/3cemp.2022.110250.164-171
169
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Example 1. Let
G
=
linear span
(
I,S,S∗
), where
S
is the unilateral right shift in
B
(
H
)and
I
is the identity operator. Let
A
=
C∗
(
G
)be the
C∗
-algebra generated by
G
. We have
K
(
H
)
⊆A
,
A/K
(
H
)
∼
=C
(
T
)is commutative, where
T
denotes the unit circle in
C
. Let
Id
denotes the identity
representation of the
C∗
-algebra
A
. Let
S∗Id
(
·
)
S
be a completely positive map on the
C∗
-algebra
A
such
that
S∗IdS|G
=
Id|G
, it is easy to see that
S∗IdS|A
=
Id|A
. Therefore the unital representation
Id|G
does not have unique extension property. Using [15, Theorem 3.1], we conclude that
G
is not a weak
hyperrigid operator system in a W∗-algebra B(H).
Let
G1
=
G
,
A1
=
A
and
Id1
denotes the identity representation of
A1
. Let
G2
=
A2
=
Mn
(
C
)and
Id2
denotes the identity representation of the
C∗
-algebra
A2
. The completely positive map
S∗Id1S⊗Id2
on the
C∗
-algebra
A1⊗A2
is such that
S∗Id1S⊗Id2
=
Id1⊗Id2
on operator system
G1⊗G2
. By
the above conclusion we see that
S∗Id1S⊗Id2
=
Id1⊗Id2
on the
C∗
-algebra
A1⊗A2
. Therefore
the unital representation
Id1⊗Id2
does not have unique extension property for
G1⊗G2
. Hence
by theorem [15, Theorem 3.1],
G1⊗G2
is not a weak hyperrigid operator system in a
W∗
-algebra
B(H)⊗Mn(C).
Example 2. Let the Volterra integration operator
V
acting on the Hilbert space
H
=
L2
[0
,
1] be given
by
Vf(x)=x
0
f(t)dt, f ∈L2[0,1].
V
generates the
C∗
-algebra
K
=
K
(
H
)of all compact operators. Let
S
=
linear span
(
V,V ∗,V2,V2∗
)
and
S
is weak hyperrigid [4, Theorem 1.7] and [15, Theorem 3.1] in
W∗
-algebra
B
(
H
). Let
S1
=
S2
=
S
and
A1
=
A2
=
B
(
H
). We know that
S1
and
S2
are weak hyperrigid operator systems in the
W∗
-algebra
A1
and
A2
respectively. By corollary 1 we conclude that
S1⊗S2
is weak hyperrigid operator system in
the W∗-algebra A1⊗A2.
Example 3. Let
G
=
linear span
(
I,S,S∗,SS∗
), where
S
is the unilateral right shift in
B
(
H
)and
I
is the identity operator. Let
A
=
C∗
(
G
)be the
C∗
-algebra generated by the operator system
G
. We
have,
K
(
H
)
⊆A
.
A/K
(
H
)
∼
=C
(
T
)is commutative, where
T
denotes the unit circle in
C
. Since
S
is
an isometry,
G
is a weak hyperrigid operator system in the
W∗
-algebra
B
(
H
)[15, Theorem 3.1]. Let
G1
=
G
,
A1
=
B
(
H
)and
G2
=
A2
=
Mn
(
C
). It is clear that
G2
is a weak hyperrigid operator system
in the
W∗
-algebra
A2
=
Mn
(
C
). By corollary 1,
G⊗Mn
(
C
)is a weak hyperrigid operator system in
B(H)⊗Mn(C).
ACKNOWLEDGMENT
The Second author thanks the Council of Scientific & Industrial Research (CSIR) for providing a doctoral
fellowship. The third author thanks Cochin University of Science and Technology for granting the
project under Seed Money for New Research Initiatives (order No. CUSAT/PL(UGC).A1/1112/2021)
dated 09.03.2021.
REFERENCES
[1]
Arunkumar, C. S., and Vijayarajan, A. K (2022). On weak boundary representations and
quasi hyperrigidity for operator systems. J. Anal., 30, no. 3, 1219–1227.
[2] Arveson, W. B. (1969). Subalgebras of C∗-algebras. Acta Math., 123, 141-224.
[3] Arveson, W. B. (1972). Subalgebras of C∗-algebras II. Acta Math., 128, 271-308.
[4]
Arveson, W. B. (2011). The noncommutative Choquet boundary II; Hyperrigidity. Israel J. Math.,
184, 349-385.
[5]
Blackadar, B. (2006). Operator algebras Theory of
C∗
-algebras and von Neumann algebras.
Encyclopaedia of Mathematical Sciences 122, Operator Algebras and Non-commutative Geometry III,
Springer-Verlag, Berlin.
https://doi.org/10.17993/3cemp.2022.110250.164-171
[6]
Davidson, K. R., and Kennedy, M. (2021). Choquet order and hyperrigidity for function
systems. Adv. Math., 385, Paper No. 107774, 30 pp.
[7]
Guichardet, A. (1969). Tensor products of
C∗
-algebras. Aarhus University Lecture Notes Series,
No. 12, Aahrus.
[8]
Hopenwasser, A. (1973). Boundary representations on
C∗
-algebras with matrix units. Trans.
Amer. Math. Soc., 177, 483-490.
[9]
Hopenwasser, A. (1978). Boundary representations and tensor products of
C∗
-algebras. Proc.
Amer. Math. Soc., 71, no. 1, 95-98.
[10]
Kleski, C. (2014). Korovkin-type properties for completely positive maps. Illinois J. Math., 58,
no. 4, 1107-1116.
[11] Korovkin, P. P. (1960). Linear operators and approximation theory. Hindustan, Delhi.
[12]
Muhly, P., and Solel, B. (1998). An algebraic characterization of boundary representations,
Nonselfadjoint operator algebras, operator theory, and related topics. Oper. Theory Adv. Appl. 104,
Birkhauser, Basel, 189-196.
[13]
Limaye, B. V., and Namboodiri, M. N. N. (1984). Weak Korovkin approximation by completely
positive linear maps on B(H).J. Approx. Theory, 42, no. 3, 201-211.
[14]
Namboodiri, M. N. N. (2011). Developments in noncommutative Korovkin theorems. RIMS
Kokyuroku Series 737, 91-104.
[15]
Namboodiri, M. N. N. (2012). Geometric theory of weak Korovkin sets. Oper. Matrices, 6, no.
2, 271-278.
[16]
Namboodiri, M. N. N.,Pramod, S.,Shankar, P., and Vijayarajan, A. K. (2018). Quasi
hyperrigidity and weak peak points for non-commutative operator systems. Proc. Indian Acad. Sci.
Math. Sci., 128, no. 5, Paper No. 66.
[17] Sakai, S. (1998). C∗-algebras and W∗-algebras. Springer-Verlag, Berlin.
[18]
Saskin, Y. A. (1966). Korovkin systems in spaces of continuous functions. Amer. Math. Soc.
Transl., 54, no. 2, 125-144.
[19] Shankar, P. (2020). Hyperrigid generators in C∗-algebras. J. Anal., 28 no. 3, 791–797.
[20]
Shankar, P., and Vijayarajan, A. K. (2017). Hyperrigid operator systems and Hilbert modules.
Ann. Funct. Anal., 8, no. 1, 133-141.
[21]
Shankar, P., and Vijayarajan, A. K. (2018). Tensor products of hyperrigid operator systems.
Ann. Funct. Anal., 9, no. 3, 369–375.
[22]
Shankar, P., and Vijayarajan, A. K. (2020). Hilbert modules characterization for weak
hyperrigid operator systems. J. Anal., 28, no. 4, 905–912.
https://doi.org/10.17993/3cemp.2022.110250.164-171
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
170
Example 1. Let
G
=
linear span
(
I,S,S∗
), where
S
is the unilateral right shift in
B
(
H
)and
I
is the identity operator. Let
A
=
C∗
(
G
)be the
C∗
-algebra generated by
G
. We have
K
(
H
)
⊆A
,
A/K
(
H
)
∼
=C
(
T
)is commutative, where
T
denotes the unit circle in
C
. Let
Id
denotes the identity
representation of the
C∗
-algebra
A
. Let
S∗Id
(
·
)
S
be a completely positive map on the
C∗
-algebra
A
such
that
S∗IdS|G
=
Id|G
, it is easy to see that
S∗IdS|A
=
Id|A
. Therefore the unital representation
Id|G
does not have unique extension property. Using [15, Theorem 3.1], we conclude that
G
is not a weak
hyperrigid operator system in a W∗-algebra B(H).
Let
G1
=
G
,
A1
=
A
and
Id1
denotes the identity representation of
A1
. Let
G2
=
A2
=
Mn
(
C
)and
Id2
denotes the identity representation of the
C∗
-algebra
A2
. The completely positive map
S∗Id1S⊗Id2
on the
C∗
-algebra
A1⊗A2
is such that
S∗Id1S⊗Id2
=
Id1⊗Id2
on operator system
G1⊗G2
. By
the above conclusion we see that
S∗Id1S⊗Id2
=
Id1⊗Id2
on the
C∗
-algebra
A1⊗A2
. Therefore
the unital representation
Id1⊗Id2
does not have unique extension property for
G1⊗G2
. Hence
by theorem [15, Theorem 3.1],
G1⊗G2
is not a weak hyperrigid operator system in a
W∗
-algebra
B(H)⊗Mn(C).
Example 2. Let the Volterra integration operator
V
acting on the Hilbert space
H
=
L2
[0
,
1] be given
by
Vf(x)=x
0
f(t)dt, f ∈L2[0,1].
V
generates the
C∗
-algebra
K
=
K
(
H
)of all compact operators. Let
S
=
linear span
(
V,V ∗,V2,V2∗
)
and
S
is weak hyperrigid [4, Theorem 1.7] and [15, Theorem 3.1] in
W∗
-algebra
B
(
H
). Let
S1
=
S2
=
S
and
A1
=
A2
=
B
(
H
). We know that
S1
and
S2
are weak hyperrigid operator systems in the
W∗
-algebra
A1
and
A2
respectively. By corollary 1 we conclude that
S1⊗S2
is weak hyperrigid operator system in
the W∗-algebra A1⊗A2.
Example 3. Let
G
=
linear span
(
I,S,S∗,SS∗
), where
S
is the unilateral right shift in
B
(
H
)and
I
is the identity operator. Let
A
=
C∗
(
G
)be the
C∗
-algebra generated by the operator system
G
. We
have,
K
(
H
)
⊆A
.
A/K
(
H
)
∼
=C
(
T
)is commutative, where
T
denotes the unit circle in
C
. Since
S
is
an isometry,
G
is a weak hyperrigid operator system in the
W∗
-algebra
B
(
H
)[15, Theorem 3.1]. Let
G1
=
G
,
A1
=
B
(
H
)and
G2
=
A2
=
Mn
(
C
). It is clear that
G2
is a weak hyperrigid operator system
in the
W∗
-algebra
A2
=
Mn
(
C
). By corollary 1,
G⊗Mn
(
C
)is a weak hyperrigid operator system in
B(H)⊗Mn(C).
ACKNOWLEDGMENT
The Second author thanks the Council of Scientific & Industrial Research (CSIR) for providing a doctoral
fellowship. The third author thanks Cochin University of Science and Technology for granting the
project under Seed Money for New Research Initiatives (order No. CUSAT/PL(UGC).A1/1112/2021)
dated 09.03.2021.
REFERENCES
[1]
Arunkumar, C. S., and Vijayarajan, A. K (2022). On weak boundary representations and
quasi hyperrigidity for operator systems. J. Anal., 30, no. 3, 1219–1227.
[2] Arveson, W. B. (1969). Subalgebras of C∗-algebras. Acta Math., 123, 141-224.
[3] Arveson, W. B. (1972). Subalgebras of C∗-algebras II. Acta Math., 128, 271-308.
[4]
Arveson, W. B. (2011). The noncommutative Choquet boundary II; Hyperrigidity. Israel J. Math.,
184, 349-385.
[5]
Blackadar, B. (2006). Operator algebras Theory of
C∗
-algebras and von Neumann algebras.
Encyclopaedia of Mathematical Sciences 122, Operator Algebras and Non-commutative Geometry III,
Springer-Verlag, Berlin.
https://doi.org/10.17993/3cemp.2022.110250.164-171
[6]
Davidson, K. R., and Kennedy, M. (2021). Choquet order and hyperrigidity for function
systems. Adv. Math., 385, Paper No. 107774, 30 pp.
[7]
Guichardet, A. (1969). Tensor products of
C∗
-algebras. Aarhus University Lecture Notes Series,
No. 12, Aahrus.
[8]
Hopenwasser, A. (1973). Boundary representations on
C∗
-algebras with matrix units. Trans.
Amer. Math. Soc., 177, 483-490.
[9]
Hopenwasser, A. (1978). Boundary representations and tensor products of
C∗
-algebras. Proc.
Amer. Math. Soc., 71, no. 1, 95-98.
[10]
Kleski, C. (2014). Korovkin-type properties for completely positive maps. Illinois J. Math., 58,
no. 4, 1107-1116.
[11] Korovkin, P. P. (1960). Linear operators and approximation theory. Hindustan, Delhi.
[12]
Muhly, P., and Solel, B. (1998). An algebraic characterization of boundary representations,
Nonselfadjoint operator algebras, operator theory, and related topics. Oper. Theory Adv. Appl. 104,
Birkhauser, Basel, 189-196.
[13]
Limaye, B. V., and Namboodiri, M. N. N. (1984). Weak Korovkin approximation by completely
positive linear maps on B(H).J. Approx. Theory, 42, no. 3, 201-211.
[14]
Namboodiri, M. N. N. (2011). Developments in noncommutative Korovkin theorems. RIMS
Kokyuroku Series 737, 91-104.
[15]
Namboodiri, M. N. N. (2012). Geometric theory of weak Korovkin sets. Oper. Matrices, 6, no.
2, 271-278.
[16]
Namboodiri, M. N. N.,Pramod, S.,Shankar, P., and Vijayarajan, A. K. (2018). Quasi
hyperrigidity and weak peak points for non-commutative operator systems. Proc. Indian Acad. Sci.
Math. Sci., 128, no. 5, Paper No. 66.
[17] Sakai, S. (1998). C∗-algebras and W∗-algebras. Springer-Verlag, Berlin.
[18]
Saskin, Y. A. (1966). Korovkin systems in spaces of continuous functions. Amer. Math. Soc.
Transl., 54, no. 2, 125-144.
[19] Shankar, P. (2020). Hyperrigid generators in C∗-algebras. J. Anal., 28 no. 3, 791–797.
[20]
Shankar, P., and Vijayarajan, A. K. (2017). Hyperrigid operator systems and Hilbert modules.
Ann. Funct. Anal., 8, no. 1, 133-141.
[21]
Shankar, P., and Vijayarajan, A. K. (2018). Tensor products of hyperrigid operator systems.
Ann. Funct. Anal., 9, no. 3, 369–375.
[22]
Shankar, P., and Vijayarajan, A. K. (2020). Hilbert modules characterization for weak
hyperrigid operator systems. J. Anal., 28, no. 4, 905–912.
https://doi.org/10.17993/3cemp.2022.110250.164-171
171
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
