Z
-HYPERRIGIDITY AND
Z
-BOUNDARY REPRESENTA-
TIONS
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: 18/10/2022 Acceptance: 02/11/2022 Publication: 29/12/2022
Suggested citation:
V. A. Anjali, Athul Augustine and P. Shankar (2022).
Z
-Hyperrigidity and
Z
-boundary representations 3C Empresa.
Investigación y pensamiento crítico,11 (2), 173-184. https://doi.org/10.17993/3cemp.2022.110250.173-184
https://doi.org/10.17993/3cemp.2022.110250.173-184
173
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
ABSTRACT
In this article, we introduce the notions of
Z
-finite representations and
Z
-separation property of
representations for operator
Z
-systems generating
C∗
-algebras. We use these notions to characterize the
Z
-boundary representations for operator
Z
-systems. We introduce
Z
-hyperrigidity of operator
Z
-systems.
We investigate an analogue version of Saskin’s theorem in the setting of operator
Z
-systems generating
C∗-algebras.
KEYWORDS
Completely positive maps, Operator systems, Representations of C∗-algebras, Hilbert C∗-modules.
2020 Mathematics Subject Classification: Primary 46L07, 47L07; Secondary 46L05.
https://doi.org/10.17993/3cemp.2022.110250.173-184
1 INTRODUCTION
Let
S
be a subspace (subalgebra) of
C
(
X
), the set of continuous functions on compact metric space
X
. The Choquet boundary of
S
consists of the points
x∈X
with the property that there is a unique
probability measure
µ
on
X
, such that
f
(
x
)=
Xfdµ
,
f∈S
. In other words, the points
x∈X
lie in
the Choquet boundary of
S
if the point evaluation functional
f→ f
(
x
),
f∈S
extends to a unique
state on the
C∗
-algebra
C
(
X
). The Choquet boundary is a significant object to study for at least
two reasons. The Choquet boundary of
S
is dense in the Shilov boundary of
S
. Shilov boundary is
the smallest closed subset of Xon which every function in Sattains its maximum modulus. Choquet
boundary supplies a tool to identify the “minimal” representations of the elements of
S
as functions on
some compact metric space. For more details, refer to [5].
Korovkin theorem [14] deals with the convergence of positive linear maps on function algebras. The
classical Korovkin theorem is as follows: for each
n∈N
, let
ϕn
:
C
[0
,
1]
→C
[0
,
1] be a positive linear
map. If
limn→∞ ||ϕn
(
f
)
−f||
for every
f∈{
1
, x, x2}
, then
limn→∞ ||ϕn
(
f
)
−f||
for every
f∈C
[0
,
1].
The set
{
1
, x, x2}
is called a Korovkin set in
C
[0
,
1]. There is a close connection between Korovkin sets
and Choquet boundaries. Saskin [5,23] proved that
G
is a Korovkin set in
C
[0
,
1] if and only if the
Choquet boundary of Gis [0,1].
Arveson [2] initiated a non-commutative analogue of the Choquet boundary in the context of unital
operator algebras and operator systems in
C∗
-algebra. The central objects in his approach are the
so-called boundary representations. Certain unital completely positive linear maps have unique extension
property, almost in the spirit of defining property for points to lie in the classical Choquet boundary. The
conjecture of Arveson states that every operator system and every unital operator algebra has sufficiently
many boundary representations to norm it completely. Hamana [11] constructed the
C∗
-envelope of the
operator system using a different method. Arveson [3] proved the conjecture for separable
C∗
-algebras.
Davidson and Kennedy [8] completely settled conjecture on boundary representations. Fuller, Hartz,
and Lupini [10] introduced the notion of boundary representations for operator spaces in ternary rings
of operators. They established the natural operator space analogue of Arveson’s conjecture on boundary
representations. Magajna [17] introduced
Z
-boundary representations for operator
Z
-system generating
a
C∗
-algebra on self-dual Hilbert
Z
-modules, where
Z
is abelian von Neumann algebra. Magajna [17]
proved analogue of Arveson’s conjecture for
Z
-boundary representations of
C∗
-algebra generated by
operator Z-systems on self-dual Hilbert Z-modules over abelian von Neumann algebra Z.
Arveson [4] introduced the notion of hyperrigid set, which is a non-commutative analogue of the
Korovkin set. Arveson studied hyperrigidity in the setting of operator systems in
C∗
-algebras, and he
tried to prove an analogue version of Saskin’s theorem using hyperrigidity and boundary representations.
Arveson [4] proved if every operator system is hyperrigid in generating
C∗
-algebra, then every irreducible
representation of
C∗
-algebra is a boundary representation for the operator system. But he could not be
able to prove the converse in generality. The converse of the above result is called Arveson’s hyperrigidity
conjecture. Hyperrigidity conjecture is as follows: for an operator system
S
and the generated
C∗
-algebra
A
, if every irreducible representation of
A
is a boundary representation for
S
, then an operator system
S
is hyperrigid. Arveson [4] showed that the hyperrigidity conjecture is valid for
C∗
-algebras with a
countable spectrum.
Davidson and Kennedy [9] established a dilation-theoretic characterization of the Choquet order
on the space of measures on a compact convex set using ideas from the theory of operator algebras.
This yields an extension of Cartier’s dilation theorem to the non-separable case and a non-separable
version of Šaškin’s theorem from approximation theory. They showed that a slight variant of this
order characterizes the representations of commutative
C∗
-algebras with the unique extension property
relative to a set of generators. This reduces the commutative case of Arveson’s hyperrigidity conjecture to
whether measures that are maximal concerning the classical Choquet order are also maximal concerning
this new order.
Kleski [13] established the hyperrigidity conjecture for all type-I
C∗
-algebras with additional
assumptions on the co-domain. The hyperrigidity conjecture is still open for general
C∗
-algebras. The
hyperrigidity conjecture inspired several studies in recent years [6,7,12,21]. Arunkumar, Shankar, and
https://doi.org/10.17993/3cemp.2022.110250.173-184
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
174
ABSTRACT
In this article, we introduce the notions of
Z
-finite representations and
Z
-separation property of
representations for operator
Z
-systems generating
C∗
-algebras. We use these notions to characterize the
Z
-boundary representations for operator
Z
-systems. We introduce
Z
-hyperrigidity of operator
Z
-systems.
We investigate an analogue version of Saskin’s theorem in the setting of operator
Z
-systems generating
C∗-algebras.
KEYWORDS
Completely positive maps, Operator systems, Representations of C∗-algebras, Hilbert C∗-modules.
2020 Mathematics Subject Classification: Primary 46L07, 47L07; Secondary 46L05.
https://doi.org/10.17993/3cemp.2022.110250.173-184
1 INTRODUCTION
Let
S
be a subspace (subalgebra) of
C
(
X
), the set of continuous functions on compact metric space
X
. The Choquet boundary of
S
consists of the points
x∈X
with the property that there is a unique
probability measure
µ
on
X
, such that
f
(
x
)=
Xfdµ
,
f∈S
. In other words, the points
x∈X
lie in
the Choquet boundary of
S
if the point evaluation functional
f→ f
(
x
),
f∈S
extends to a unique
state on the
C∗
-algebra
C
(
X
). The Choquet boundary is a significant object to study for at least
two reasons. The Choquet boundary of
S
is dense in the Shilov boundary of
S
. Shilov boundary is
the smallest closed subset of Xon which every function in Sattains its maximum modulus. Choquet
boundary supplies a tool to identify the “minimal” representations of the elements of
S
as functions on
some compact metric space. For more details, refer to [5].
Korovkin theorem [14] deals with the convergence of positive linear maps on function algebras. The
classical Korovkin theorem is as follows: for each
n∈N
, let
ϕn
:
C
[0
,
1]
→C
[0
,
1] be a positive linear
map. If
limn→∞ ||ϕn
(
f
)
−f||
for every
f∈{
1
, x, x2}
, then
limn→∞ ||ϕn
(
f
)
−f||
for every
f∈C
[0
,
1].
The set
{
1
, x, x2}
is called a Korovkin set in
C
[0
,
1]. There is a close connection between Korovkin sets
and Choquet boundaries. Saskin [5,23] proved that
G
is a Korovkin set in
C
[0
,
1] if and only if the
Choquet boundary of Gis [0,1].
Arveson [2] initiated a non-commutative analogue of the Choquet boundary in the context of unital
operator algebras and operator systems in
C∗
-algebra. The central objects in his approach are the
so-called boundary representations. Certain unital completely positive linear maps have unique extension
property, almost in the spirit of defining property for points to lie in the classical Choquet boundary. The
conjecture of Arveson states that every operator system and every unital operator algebra has sufficiently
many boundary representations to norm it completely. Hamana [11] constructed the
C∗
-envelope of the
operator system using a different method. Arveson [3] proved the conjecture for separable
C∗
-algebras.
Davidson and Kennedy [8] completely settled conjecture on boundary representations. Fuller, Hartz,
and Lupini [10] introduced the notion of boundary representations for operator spaces in ternary rings
of operators. They established the natural operator space analogue of Arveson’s conjecture on boundary
representations. Magajna [17] introduced
Z
-boundary representations for operator
Z
-system generating
a
C∗
-algebra on self-dual Hilbert
Z
-modules, where
Z
is abelian von Neumann algebra. Magajna [17]
proved analogue of Arveson’s conjecture for
Z
-boundary representations of
C∗
-algebra generated by
operator Z-systems on self-dual Hilbert Z-modules over abelian von Neumann algebra Z.
Arveson [4] introduced the notion of hyperrigid set, which is a non-commutative analogue of the
Korovkin set. Arveson studied hyperrigidity in the setting of operator systems in
C∗
-algebras, and he
tried to prove an analogue version of Saskin’s theorem using hyperrigidity and boundary representations.
Arveson [4] proved if every operator system is hyperrigid in generating
C∗
-algebra, then every irreducible
representation of
C∗
-algebra is a boundary representation for the operator system. But he could not be
able to prove the converse in generality. The converse of the above result is called Arveson’s hyperrigidity
conjecture. Hyperrigidity conjecture is as follows: for an operator system
S
and the generated
C∗
-algebra
A
, if every irreducible representation of
A
is a boundary representation for
S
, then an operator system
S
is hyperrigid. Arveson [4] showed that the hyperrigidity conjecture is valid for
C∗
-algebras with a
countable spectrum.
Davidson and Kennedy [9] established a dilation-theoretic characterization of the Choquet order
on the space of measures on a compact convex set using ideas from the theory of operator algebras.
This yields an extension of Cartier’s dilation theorem to the non-separable case and a non-separable
version of Šaškin’s theorem from approximation theory. They showed that a slight variant of this
order characterizes the representations of commutative
C∗
-algebras with the unique extension property
relative to a set of generators. This reduces the commutative case of Arveson’s hyperrigidity conjecture to
whether measures that are maximal concerning the classical Choquet order are also maximal concerning
this new order.
Kleski [13] established the hyperrigidity conjecture for all type-I
C∗
-algebras with additional
assumptions on the co-domain. The hyperrigidity conjecture is still open for general
C∗
-algebras. The
hyperrigidity conjecture inspired several studies in recent years [6,7,12,21]. Arunkumar, Shankar, and
https://doi.org/10.17993/3cemp.2022.110250.173-184
175
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Vijayarajan [1] introduced rectangular hyperrigidity in setting operator spaces in a ternary ring of
operators. They established an analogue version of Saskin’s theorem in the case of a finite-dimensional
ternary ring of operators, and they gave some partial answers analogue to results in the papers [4,13].
This paper is divided into three sections besides the introduction. In Section 2, we gather the
necessary background material and required results. In section 3, we introduce the notions of
Z
-finite
representations for operator
Z
-systems and
Z
-separation property of operator
Z
-systems. These notions
are generalizations of finite representation and separating property for representations introduced
by Arveson [2]. We use these notions to characterize the
Z
-boundary representations for operator
Z
-systems. In section 4, We introduce
Z
-hyperrigidity of operator
Z
-systems in generating
C∗
-algebras
which is a generalization of hyperrigidity introduced by Arveson [4]. We investigate an analogue version
of Saskin’s theorem in the setting of operator Z-systems generating C∗-algebra.
2 PRELIMINARIES
Arepresentation of a unital
C∗
-algebra
A
on a Hilbert space
H
makes
H
a Hilbert
A
-module. Let
BA
(
H
)denote the set of all bounded
A
-module maps on
H
. We will denote von Neumann algebras by
A,B, ..., Z
and general
C∗
-algebras by
A, B, ....
Let
A
be
C∗
-algebra and
A
is faithfully represented
on a Hilbert space
H
.
X⊆B
(
H
)is said to be a faithful operator
C
-system if
X
is a norm closed
self-adjoint C-subbimodule of B(H)(for more details and abstract characterization refer [22]).
Let
H
be a Hilbert
A
-module. Let
CCPA
(
X, B
(
H
)) denote the set of all contractive completely
positive
A
-bimodule maps form
X
into
B
(
H
). Let
UCPA
(
X, B
(
H
)) denote the set of all unital completely
positive
A
-bimodule maps form
X
into
B
(
H
). Let
X
be a faithful operator
A
-system contained in
a
C∗
-algebra
B
so that
A
and
B
have the same unit 1. By the well-known multiplicative domain
argument [22, 3.18] any completely positive extension to
B
of a map
φ∈UCPA
(
X, B
(
H
)) must be a
A-bimodule map since φextends the representation φ|A.
The motive of this article is to extend the main results of the papers [2], and [4] in the context of
Hilbert spaces are replaced by Hilbert
C∗
-modules over abelian von Neumann algebra
Z
. For a theory
of Hilbert
C∗
-modules, we refer to [15,19]. Hilbert
C∗
-modules over von Neumann algebras
Z
are like
Hilbert spaces, except that the inner product takes values in
Z
. Let
E
be Hilbert
Z
-module, we denote
⟨·,·⟩
the
Z
-valued inner product on
E
and let
|e|
:=
⟨e, e⟩
the corresponding
Z
-valued norm. For
e∈E
, the scalar-valued norm is denoted by
||e||
:=
||⟨x, x⟩||1
2
. A Hilbert
Z
-module is said to self-dual
if each
Z
-module map
ϕ
from
E
to
Z
has the form
ϕ
(
e
)=
⟨e, f⟩
for an
f∈E
. Let
BZ
(
E
)denote the
set of all bounded
Z
-module endomorphisms of
E
. If
E
is self-dual then
BZ
(
E
)is adjointable. If
E⊆F
are self-dual C∗-modules over Zthen F=E⊕E⊥.
The following definitions and results are due to Magajna [17]. A map
ψ∈UCPZ
(
X, BZ
(
F
)) is called
Z
- dilation of
φ∈UCPZ
(
X, BZ
(
E
)) for self-dual
C∗
-module
F⊇E
over
Z
if
pψ
(
x
)
|E
=
φ
(
x
)
∀x∈X
,
where
p
:
F→E
is the orthogonal projection. We write
ψ⪰Zφ
if
ψ
is a
Z
-dilation of
φ
. A map
φ∈UCPZ
(
X, BZ
(
E
)) is said to be
Z
-maximal if every
ψ∈UCPZ
(
X, BZ
(
F
)), where
F
is a self-dual
C∗-module over Z, satisfying ψ⪰Zφ, decomposes as ψ=φ⊕θfor some θ∈UCPZ(X, BZ(E⊥)).
Remark 1. From [17, Remark 4.12] observe that, if an operator
Z
-system
X
is contained in a
C∗
-
algebra
B
generated by
X
and containing
Z
in its center, any map
φ∈UCPZ
(
X, BZ
(
E
)) can be
extended to a map ˜φ∈UCPZ(B,BZ(E)). An analogue version of Stinespring’s dilation theorem for ˜φ
can be represented as follows:
˜φ(b)=V∗π(b)V∀b∈B,
where
π
:
B→BZ
(
F
)is a representation on a self-dual
C∗
-module
F
over
Z
and
V∈BZ
(
E,F
)is
an isometry such that [
π
(
B
)
VE
]=
F
. Observe that [
π
(
B
)
VE
]=
F
is the minimality condition for
an analogue version of Stinespring’s decomposition. For more details see [15, Theorem 5.6] and [20,
Corollary 5.3]. Paschke [20, Proposition 5.4] proved the analogue of Arveson’s [2, Theorem 1.4.2] affine
order isomorphism theorem.
https://doi.org/10.17993/3cemp.2022.110250.173-184
Definition 1. [17] A map
φ∈UCPZ
(
X, BZ
(
E
)) is said to have a
Z
-unique extension property
(
Z
-u.e.p) if
φ
has a unique completely positive
Z
-bimodule extension
˜φ
:
C∗
(
X
)
→BZ
(
E
)and
˜φ
is a
representation of C∗(X)on E.
Arveson [3, Proposition 2.4] proved that maximality is equivalent to the notion of unique extension
property in the Hilbert space setting. Similar arguments from [3, Proposition 2.4] imply that the idea of
Z
-maximality is equivalent to the notion of
Z
-unique extension property in Hilbert
Z
-module setting.
A representation (i.e., a homomorphism of
C∗
-algebras)
π
:
B→BZ
(
E
)is said to be
Z
-irreducible
if π(B)′=π(Z).
Definition 2. [17] A map
φ∈UCPZ
(
X, BZ
(
E
)) is said to be
Z
-pure if every
ψ∈UCPZ
(
X, BZ
(
E
)),
ψ≤φimplies that ψ=cφ, where c∈Z.
Remark 2. We can observe that an analogue of [2, Corollary 1.4.3] follows from [17, Remark 4.12
and Remark 4.14]. A non zero pure map in
UCPZ
(
B,BZ
(
E
)) are precisely those of the form
˜φ
(
b
)=
V∗π
(
b
)
V∀b∈B
, where
π
is an
Z
-irreducible representation of
B
on some self-dual Hilbert
C∗-module Fover Zand V∈BZ(E,F),V=0.
Definition 3. [17] A
Z
-irreducible representation
π
:
B→BZ
(
E
)(for some self-dual
E
) is called
Z-boundary representation of Bfor Xif π|Xhas the Z-unique extension property.
Magajna [17] proved analogue of Arveson’s conjecture on Z-boundary representations as follows:
Theorem 1. If
X
is a central operator
Z
-system generating a
C∗
-algebra
A
, then
Z
-boundary repre-
sentation of Afor Xon self-dual Hilbert C∗-modules over Zcompletely norm X.
3 Z-BOUNDARY REPRESENTATION
This section establishes the characterization theorem for
Z
-boundary representations. This characteriza-
tion theorem is an analogue version of [2, Theorem 2.4.5]. In general, checking the given representation
is
Z
-boundary representation is not easy. Using this characterization theorem, at least we can detect
the representations that are not Z-boundary representations.
Proposition 1. Let
X
be a operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. If
π
is a
Z-boundary representation of Bfor Xthen π|Xis Z-pure.
Proof. Let
E
self-dual Hilbert
C∗
-module over
Z
on which
π
acts. Let
φ1,φ
2∈CPZ
(
X, BZ
(
E
)) be
such that
π|X
=
φ1
+
φ2
. By [17, Remark 4.12] each
φi
can be extended to unital completely positive
Z
-bimodule map
˜φi
:
B→BZ
(
E
)such that
˜φi|X
=
φ
for
i
=1
,
2. Observe that
˜φ1
+
˜φ2
:
B→BZ
(
E
)
is a completely positive
Z
-bimodule extension of
π|X
. Since
π
is a
Z
-boundary representation for
X
,
thus
˜φ1
(
b
)+
˜φ2
(
b
)=
π
(
b
)for all
b∈B
. Also,
π
is an
Z
-irreducible representation of
B
so by Remark
2
˜φ1
+
˜φ2
is a
Z
-pure map in
CPZ
(
B,BZ
(
E
)). Thus, there are
ci∈Z
such that
˜φi
=
ciπ
on
B
for
i=1,2. Restricting to Xwe have φi=ciπ|Xfor i=1,2. Hence π|Xis Z-pure.
Magajna [16,18] studied an analogue of
C∗
-convexity and
C∗
-extreme points of operators on Hilbert
C∗
-modules. He introduced
A
-convexity and
A
-extreme points as follows: Let
K
be a Hilbert module
over a
C∗
-algebra
A
. A subset
K⊆BA
(
K
)is called
A
-convex if
n
j=1 a∗
jyjaj∈K
whenever
yj∈K
,
aj∈A
and
n
j=1 a∗
jaj
=1. A point
x
in an
A
-convex set
K
is called an
A
-extreme point of
K
if the
condition
x
=
n
j=1 a∗
jyjaj
, where
xj∈K
,
aj∈A
,
n
j=1 a∗
jaj
=1(n finite) and
aj
are invertible,
implies that there exist unitary elements
uj∈A
such that
xj
=
u∗
jxuj
. By [16, Lemma 5.5], it is enough
to check the A-extreme point condition for the case n=2.
Proposition 2. Let
X
be a operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. Let
π
be
a
Z
-irreducible representation of
B
such that
K
=
{φ∈UCPZ
(
B,BZ
(
E
)) :
φ|X
=
π|X}
. If
π
is a
Z-boundary representation of Bfor Xthen every φ∈Kis a Z-extreme point of K.
https://doi.org/10.17993/3cemp.2022.110250.173-184
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
176
Vijayarajan [1] introduced rectangular hyperrigidity in setting operator spaces in a ternary ring of
operators. They established an analogue version of Saskin’s theorem in the case of a finite-dimensional
ternary ring of operators, and they gave some partial answers analogue to results in the papers [4,13].
This paper is divided into three sections besides the introduction. In Section 2, we gather the
necessary background material and required results. In section 3, we introduce the notions of
Z
-finite
representations for operator
Z
-systems and
Z
-separation property of operator
Z
-systems. These notions
are generalizations of finite representation and separating property for representations introduced
by Arveson [2]. We use these notions to characterize the
Z
-boundary representations for operator
Z
-systems. In section 4, We introduce
Z
-hyperrigidity of operator
Z
-systems in generating
C∗
-algebras
which is a generalization of hyperrigidity introduced by Arveson [4]. We investigate an analogue version
of Saskin’s theorem in the setting of operator Z-systems generating C∗-algebra.
2 PRELIMINARIES
Arepresentation of a unital
C∗
-algebra
A
on a Hilbert space
H
makes
H
a Hilbert
A
-module. Let
BA
(
H
)denote the set of all bounded
A
-module maps on
H
. We will denote von Neumann algebras by
A,B, ..., Z
and general
C∗
-algebras by
A, B, ....
Let
A
be
C∗
-algebra and
A
is faithfully represented
on a Hilbert space
H
.
X⊆B
(
H
)is said to be a faithful operator
C
-system if
X
is a norm closed
self-adjoint C-subbimodule of B(H)(for more details and abstract characterization refer [22]).
Let
H
be a Hilbert
A
-module. Let
CCPA
(
X, B
(
H
)) denote the set of all contractive completely
positive
A
-bimodule maps form
X
into
B
(
H
). Let
UCPA
(
X, B
(
H
)) denote the set of all unital completely
positive
A
-bimodule maps form
X
into
B
(
H
). Let
X
be a faithful operator
A
-system contained in
a
C∗
-algebra
B
so that
A
and
B
have the same unit 1. By the well-known multiplicative domain
argument [22, 3.18] any completely positive extension to
B
of a map
φ∈UCPA
(
X, B
(
H
)) must be a
A-bimodule map since φextends the representation φ|A.
The motive of this article is to extend the main results of the papers [2], and [4] in the context of
Hilbert spaces are replaced by Hilbert
C∗
-modules over abelian von Neumann algebra
Z
. For a theory
of Hilbert
C∗
-modules, we refer to [15,19]. Hilbert
C∗
-modules over von Neumann algebras
Z
are like
Hilbert spaces, except that the inner product takes values in
Z
. Let
E
be Hilbert
Z
-module, we denote
⟨·,·⟩
the
Z
-valued inner product on
E
and let
|e|
:=
⟨e, e⟩
the corresponding
Z
-valued norm. For
e∈E
, the scalar-valued norm is denoted by
||e||
:=
||⟨x, x⟩||1
2
. A Hilbert
Z
-module is said to self-dual
if each
Z
-module map
ϕ
from
E
to
Z
has the form
ϕ
(
e
)=
⟨e, f⟩
for an
f∈E
. Let
BZ
(
E
)denote the
set of all bounded
Z
-module endomorphisms of
E
. If
E
is self-dual then
BZ
(
E
)is adjointable. If
E⊆F
are self-dual C∗-modules over Zthen F=E⊕E⊥.
The following definitions and results are due to Magajna [17]. A map
ψ∈UCPZ
(
X, BZ
(
F
)) is called
Z
- dilation of
φ∈UCPZ
(
X, BZ
(
E
)) for self-dual
C∗
-module
F⊇E
over
Z
if
pψ
(
x
)
|E
=
φ
(
x
)
∀x∈X
,
where
p
:
F→E
is the orthogonal projection. We write
ψ⪰Zφ
if
ψ
is a
Z
-dilation of
φ
. A map
φ∈UCPZ
(
X, BZ
(
E
)) is said to be
Z
-maximal if every
ψ∈UCPZ
(
X, BZ
(
F
)), where
F
is a self-dual
C∗-module over Z, satisfying ψ⪰Zφ, decomposes as ψ=φ⊕θfor some θ∈UCPZ(X, BZ(E⊥)).
Remark 1. From [17, Remark 4.12] observe that, if an operator
Z
-system
X
is contained in a
C∗
-
algebra
B
generated by
X
and containing
Z
in its center, any map
φ∈UCPZ
(
X, BZ
(
E
)) can be
extended to a map ˜φ∈UCPZ(B,BZ(E)). An analogue version of Stinespring’s dilation theorem for ˜φ
can be represented as follows:
˜φ(b)=V∗π(b)V∀b∈B,
where
π
:
B→BZ
(
F
)is a representation on a self-dual
C∗
-module
F
over
Z
and
V∈BZ
(
E,F
)is
an isometry such that [
π
(
B
)
VE
]=
F
. Observe that [
π
(
B
)
VE
]=
F
is the minimality condition for
an analogue version of Stinespring’s decomposition. For more details see [15, Theorem 5.6] and [20,
Corollary 5.3]. Paschke [20, Proposition 5.4] proved the analogue of Arveson’s [2, Theorem 1.4.2] affine
order isomorphism theorem.
https://doi.org/10.17993/3cemp.2022.110250.173-184
Definition 1. [17] A map
φ∈UCPZ
(
X, BZ
(
E
)) is said to have a
Z
-unique extension property
(
Z
-u.e.p) if
φ
has a unique completely positive
Z
-bimodule extension
˜φ
:
C∗
(
X
)
→BZ
(
E
)and
˜φ
is a
representation of C∗(X)on E.
Arveson [3, Proposition 2.4] proved that maximality is equivalent to the notion of unique extension
property in the Hilbert space setting. Similar arguments from [3, Proposition 2.4] imply that the idea of
Z
-maximality is equivalent to the notion of
Z
-unique extension property in Hilbert
Z
-module setting.
A representation (i.e., a homomorphism of
C∗
-algebras)
π
:
B→BZ
(
E
)is said to be
Z
-irreducible
if π(B)′=π(Z).
Definition 2. [17] A map
φ∈UCPZ
(
X, BZ
(
E
)) is said to be
Z
-pure if every
ψ∈UCPZ
(
X, BZ
(
E
)),
ψ≤φimplies that ψ=cφ, where c∈Z.
Remark 2. We can observe that an analogue of [2, Corollary 1.4.3] follows from [17, Remark 4.12
and Remark 4.14]. A non zero pure map in
UCPZ
(
B,BZ
(
E
)) are precisely those of the form
˜φ
(
b
)=
V∗π
(
b
)
V∀b∈B
, where
π
is an
Z
-irreducible representation of
B
on some self-dual Hilbert
C∗-module Fover Zand V∈BZ(E,F),V=0.
Definition 3. [17] A
Z
-irreducible representation
π
:
B→BZ
(
E
)(for some self-dual
E
) is called
Z-boundary representation of Bfor Xif π|Xhas the Z-unique extension property.
Magajna [17] proved analogue of Arveson’s conjecture on Z-boundary representations as follows:
Theorem 1. If
X
is a central operator
Z
-system generating a
C∗
-algebra
A
, then
Z
-boundary repre-
sentation of Afor Xon self-dual Hilbert C∗-modules over Zcompletely norm X.
3 Z-BOUNDARY REPRESENTATION
This section establishes the characterization theorem for
Z
-boundary representations. This characteriza-
tion theorem is an analogue version of [2, Theorem 2.4.5]. In general, checking the given representation
is
Z
-boundary representation is not easy. Using this characterization theorem, at least we can detect
the representations that are not Z-boundary representations.
Proposition 1. Let
X
be a operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. If
π
is a
Z-boundary representation of Bfor Xthen π|Xis Z-pure.
Proof. Let
E
self-dual Hilbert
C∗
-module over
Z
on which
π
acts. Let
φ1,φ
2∈CPZ
(
X, BZ
(
E
)) be
such that
π|X
=
φ1
+
φ2
. By [17, Remark 4.12] each
φi
can be extended to unital completely positive
Z
-bimodule map
˜φi
:
B→BZ
(
E
)such that
˜φi|X
=
φ
for
i
=1
,
2. Observe that
˜φ1
+
˜φ2
:
B→BZ
(
E
)
is a completely positive
Z
-bimodule extension of
π|X
. Since
π
is a
Z
-boundary representation for
X
,
thus
˜φ1
(
b
)+
˜φ2
(
b
)=
π
(
b
)for all
b∈B
. Also,
π
is an
Z
-irreducible representation of
B
so by Remark
2
˜φ1
+
˜φ2
is a
Z
-pure map in
CPZ
(
B,BZ
(
E
)). Thus, there are
ci∈Z
such that
˜φi
=
ciπ
on
B
for
i=1,2. Restricting to Xwe have φi=ciπ|Xfor i=1,2. Hence π|Xis Z-pure.
Magajna [16,18] studied an analogue of
C∗
-convexity and
C∗
-extreme points of operators on Hilbert
C∗
-modules. He introduced
A
-convexity and
A
-extreme points as follows: Let
K
be a Hilbert module
over a
C∗
-algebra
A
. A subset
K⊆BA
(
K
)is called
A
-convex if
n
j=1 a∗
jyjaj∈K
whenever
yj∈K
,
aj∈A
and
n
j=1 a∗
jaj
=1. A point
x
in an
A
-convex set
K
is called an
A
-extreme point of
K
if the
condition
x
=
n
j=1 a∗
jyjaj
, where
xj∈K
,
aj∈A
,
n
j=1 a∗
jaj
=1(n finite) and
aj
are invertible,
implies that there exist unitary elements
uj∈A
such that
xj
=
u∗
jxuj
. By [16, Lemma 5.5], it is enough
to check the A-extreme point condition for the case n=2.
Proposition 2. Let
X
be a operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. Let
π
be
a
Z
-irreducible representation of
B
such that
K
=
{φ∈UCPZ
(
B,BZ
(
E
)) :
φ|X
=
π|X}
. If
π
is a
Z-boundary representation of Bfor Xthen every φ∈Kis a Z-extreme point of K.
https://doi.org/10.17993/3cemp.2022.110250.173-184
177
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
Proof. Let
φ
be in
K
. Suppose
φ
=
n
i=1 V∗
iφiVi
, where
φi∈K
,
Vi∈Z
,
n
i=1 V∗
iVi
=1(
n
finite) and
Vi
are invertible. Since
π
is a
Z
-boundary representation of
B
for
X
and
π|X
=
φ|X
, we have
π
=
φ
on
B
. An analogue version of minimal Stinespring decomposition of
φ
is trivial. Thus, by [20, Proposition
5.4] the inequality
V∗
iφiVi≤φ
implies that there exist positive contractions
Si∈φ
(
B
)
′
=
φ
(
Z
)such
that
V∗
iφiVi
=
Siφ
. Therefore
φi
=(
S
1
2
iV−1
i
)
∗φ
(
S
1
2
iV−1
i
)and
φi
(1) =
φ
(1)=1. Thus
S
1
2
iV−1
i
is an
isometry. Again, using an analogue of minimal Stinespring decomposition of
φ
is trivial. We conclude
that φiis unitarily equivalent to φfor every i. Hence φis a Z-extreme point of K.
We introduce a Z-finite representation, an analogue version of finite representation from [2].
Definition 4. Let
X
be an operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. Let
π
:
B→BZ
(
E
)
be a representation of
B
.
π
is called
Z
-finite representation for
X
if for every isometry
V∈BZ
(
E
)the
condition V∗π(x)V=π(x)for all x∈X, implies that Vis unitary.
Proposition 3. Let
X
be an operator
Z
-system in a
C∗
-algebra
B
such that
B
=
C∗
(
X
)and let
π
be
an
Z
-irreducible representation of
B
. If
π
is a
Z
-boundary representation for
X
then
π
is a
Z
-finite
representation for X.
Proof. Let
π
acts on the self-dual Hilbert
C∗
-module
E
and let
V
be an isometry in
BZ
(
E
)such
that
V∗π
(
x
)
V
=
π
(
x
)for all
x∈X
. Then
V∗π
(
·
)
V
:
B→BZ
(
E
)is a completely positive
Z
-bimodule
extension of
π|X
. Since
π
is a
Z
-boundary representation implies that
V∗π
(
b
)
V
=
π
(
b
)for all
b∈B
.
We have
V
is isometry and [
π
(
B
)
VE
]=
E
implies
VE
is a reducing subspace for
π
(
B
). Also,
π
is
Z-irreducible implies VE=E. Therefore Vis unitary. Hence πis a Z-finite representation for X.
We introduce separating operator
Z
-system, an analogue version of separating operator system
from [2].
Definition 5. Let
X
be an operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. Let
π
:
B→
BZ
(
E
)be a
Z
-irreducible representation of
B
. We say that
XZ
-separates
π
if for every
Z
-irreducible
representation
σ
of
B
on self-dual Hilbert
Z
-module
F
and for every isometry
V∈BZ
(
E,F
), the
condition V∗σ(x)V=π(x)for all x∈Ximplies that σand πare unitarily equivalent representations
of B.
Proposition 4. Let
X
be an operator
Z
-system in a
C∗
-algebra
B
such that
B
=
C∗
(
X
). If
π
:
B→
BZ(E)is a Z-boundary representation of Bfor Xthen XZ-separates π.
Proof. Let
σ
:
B→BZ
(
F
)be a
Z
-irreducible representation of
B
, where
F
is self-dual Hilbert
Z
-module and
V∈BZ
(
E,F
)is an isometry such that
V∗σ
(
x
)
V
=
π
(
x
)for all
x∈X
. Since
π
is a
Z
-boundary representation for
X
and
V∗σ
(
·
)
V
is a completely positive
Z
-bimodule extension of
π|X
implies
V∗σ
(
b
)
V
=
π
(
b
)for all
b∈B
. We have
V
is isometry and [
π
(
B
)
VE
]=
F
implies
VE
is a
reducing subspace for
σ
(
B
). Also,
σ
is
Z
-irreducible implies
VE
=
F
. Thus
V
is unitary, showing that
σand πare unitarily equivalent representations. Hence XZ-separates π.
The characterization theorem for Z-boundary representations as follows:
Theorem 2. Let
X
be an operator
Z
-systems in a
C∗
-algebra
B
such that
B
=
C∗
(
X
). Let
π
be an
Z
-irreducible representation of
B
. Then
π
is a
Z
-boundary representation for
X
if and only if the
following conditions are satisfied:
(i) πis a Z-finite representation for X.
(ii) π|Xis Z-pure.
(iii) XZ-separates π.
(iv) Let K={φ∈CPZ(B,BZ(E)) : φ|X=π|X}and every φin Kis Z-extreme point of K.
https://doi.org/10.17993/3cemp.2022.110250.173-184
Proof. Suppose
π
is a
Z
-boundary representation of
B
for
X
then conditions (i), (ii),(iii) and (iv)
follows from Proposition 3, Proposition 1, Proposition 4 and Proposition 2.
Conversely, Suppose the
Z
-irreducible representation satisfies all four conditions (i), (ii), (iii), and
(iv). Let
K
=
{φ∈CPZ
(
B,BZ
(
E
)) :
φ|X
=
π|X}
. To show
π
is a
Z
-boundary representation for
X
, it is
enough to show that
K
is
{π}
. Using (iv), let
φ
be
Z
-extreme point of
K
. Now, we prove that
φ
is a
Z
-
pure in
UCPZ
(
B,BZ
(
E
)). Let
φ1,φ
2
be in
K
such that
φ1
(
b
)+
φ2
(
b
)=
φ
(
b
)for all
b∈B
. In particular,
φ1
(
x
)+
φ2
(
x
)=
φ
(
x
)for all
x∈X
. Our assumption,
φ|X
=
π|X
is pure implies there exists
ci≥
0in
Z
such that
φi
(
x
)=
ciφ
(
x
)for all
x∈X
. If
c1
=0and 1
∈X
then
φ1
(1) = 0, thus
φ1
=0. This contracts
to the choice of
φ1
, therefore
c1>
0and similarly
c2>
0. Using [16, Definition 5.1 and Proposition 5.2],
and every
Z
-extreme points are Choquet
Z
-points, we have
φi
=(
c
1
2
i
)
∗φ
(
c
1
2
i
)and (
c
1
2
1
)
∗c
1
2
1
+(
c
1
2
2
)
∗c
1
2
2
=1.
Now put
ψi
=(
c−1
2
i
)
∗φi
(
c−1
2
i
)for
i
=1
,
2. Then
ψi∈K
and (
c
1
2
1
)
∗ψ1
(
c
1
2
1
)+(
c
1
2
2
)
∗ψ2
(
c
1
2
1
)=
φ
. Since
φ
is
Z
-extreme point of
K
then there exists unitary elements
ui∈Z
for
i
=1
,
2such that
ψi
=
u∗
iφui
for
i
=1
,
2.Wehave(
c−1
2
i
)
∗φi
(
c−1
2
i
)=
u∗
iφui
for
i
=1
,
2and
c−1
iφi
=
u∗
iφui
for
i
=1
,
2. By [16, Definition
5.1 and Proposition 5.2], thus φi=ciφfor i=1,2. Hence φis a Z-pure in UCPZ(B,BZ(E)).
By Remark 2, there is a
Z
-irreducible representation
σ
of
B
on a self-dual Hilbert
Z
-module
F
and an isometry
V∈BZ
(
E,F
)such that
φ
=
V∗σV
. In particular,
π
(
x
)=
φ
(
x
)=
V∗σ
(
x
)
V
for all
x∈X
. The assumption (iii),
XZ
-separates
π
implies that
σ
is unitarily equivalent to
π
. Thus there
exists a unitary
U∈BZ
(
E,F
)such that
σ
=
U∗πU
. Therefore we have
π
(
x
)=(
UV
)
∗π
(
x
)
UV
for all
x∈X
.
UV
is isometry in
BZ
(
E
). The assumption (i),
π
is a
Z
-finite representation for
X
implies
UV
is unitary. Thus
V
=
U∗UV
is a unitary in
BZ
(
E,F
). Now
π|X
=
V∗σV |X
becomes
π
(
x
)=
V−1σ
(
x
)
V
for all
x∈X
.
V−1σV
is a representation of
B
which agrees with
π
on
X
. Therefore
π
(
b
)=
V−1σ
(
b
)
V
for all b∈B=C∗(X). Hence φ=πon B.
4 Z-HYPERRIGIDITY
In this section, we introduce the notion of
Z
-hyperrigidity in the operator
Z
-system.
Z
-hyperrigidity is
an analogue version of Arveson’s [4] notion of hyperrigidity. We define Z-hyperrigidity as follows:
Definition 6. Let
A
be a
C∗
-algebra, and let
G⊆A
(finite or countably infinite) be a set of generators
of
A
(i.e.,
A
=
C∗
(
G
)). Then
G
is said to be
Z
-hyperrigid if for every faithful representation
A⊆BZ
(
E
)
of
A
on a self-dual Hilbert
Z
-module
E
and every sequence of unital completely positive
Z
-bimodule
maps φn:BZ(E)→BZ(E),n=1,2, ...,
lim
n→∞ ||φn(g)−g|| =0,∀g∈G=⇒lim
n→∞ ||φn(a)−a|| =0,∀a∈A. (1)
We have lightened notation in the above definition by identifying the
C∗
-algebra
A
with its image
π
(
A
)in a faithful nondegenerate representation
π
:
A→BZ
(
E
)on a self-dual Hilbert
Z
-module
E
.
Notably,
Z
-hyperrigidity of operator
Z
-system on a self-dual Hilbert
Z
-module implies not only that
(1) should hold for sequences of UCP
Z
-bimodule maps
φn
defined on
BZ
(
E
), but also that the property
should hold for every other faithful representation of
A
. If
Z
=
C
, then the definition of
Z
-hyperrigity
is the same as the definition of hyperrigidity in [4, definition 1.1].
Proposition 5. Let Abe a C∗-algebra and Gbe a generating subset of A. Then Gis Z-hyperrigid if
and only if the operator Z-system generated by Gis Z-hyperrigid.
Proof. The proof follows directly from the definition of Z-hyperrigidity.
Now we prove the characterization theorem for Z-hyperrigid operator Z-systems.
Theorem 3. Let
X
be a separable operator
Z
-system and
X
generates a
C∗
-algebra
A
(i.e.,
A
=
C∗
(
X
)).
The following are equivalent:
(i) Xis Z-hyperrigid.
https://doi.org/10.17993/3cemp.2022.110250.173-184
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
178
Proof. Let
φ
be in
K
. Suppose
φ
=
n
i=1 V∗
iφiVi
, where
φi∈K
,
Vi∈Z
,
n
i=1 V∗
iVi
=1(
n
finite) and
Vi
are invertible. Since
π
is a
Z
-boundary representation of
B
for
X
and
π|X
=
φ|X
, we have
π
=
φ
on
B
. An analogue version of minimal Stinespring decomposition of
φ
is trivial. Thus, by [20, Proposition
5.4] the inequality
V∗
iφiVi≤φ
implies that there exist positive contractions
Si∈φ
(
B
)
′
=
φ
(
Z
)such
that
V∗
iφiVi
=
Siφ
. Therefore
φi
=(
S
1
2
iV−1
i
)
∗φ
(
S
1
2
iV−1
i
)and
φi
(1) =
φ
(1)=1. Thus
S
1
2
iV−1
i
is an
isometry. Again, using an analogue of minimal Stinespring decomposition of
φ
is trivial. We conclude
that φiis unitarily equivalent to φfor every i. Hence φis a Z-extreme point of K.
We introduce a Z-finite representation, an analogue version of finite representation from [2].
Definition 4. Let
X
be an operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. Let
π
:
B→BZ
(
E
)
be a representation of
B
.
π
is called
Z
-finite representation for
X
if for every isometry
V∈BZ
(
E
)the
condition V∗π(x)V=π(x)for all x∈X, implies that Vis unitary.
Proposition 3. Let
X
be an operator
Z
-system in a
C∗
-algebra
B
such that
B
=
C∗
(
X
)and let
π
be
an
Z
-irreducible representation of
B
. If
π
is a
Z
-boundary representation for
X
then
π
is a
Z
-finite
representation for X.
Proof. Let
π
acts on the self-dual Hilbert
C∗
-module
E
and let
V
be an isometry in
BZ
(
E
)such
that
V∗π
(
x
)
V
=
π
(
x
)for all
x∈X
. Then
V∗π
(
·
)
V
:
B→BZ
(
E
)is a completely positive
Z
-bimodule
extension of
π|X
. Since
π
is a
Z
-boundary representation implies that
V∗π
(
b
)
V
=
π
(
b
)for all
b∈B
.
We have
V
is isometry and [
π
(
B
)
VE
]=
E
implies
VE
is a reducing subspace for
π
(
B
). Also,
π
is
Z-irreducible implies VE=E. Therefore Vis unitary. Hence πis a Z-finite representation for X.
We introduce separating operator
Z
-system, an analogue version of separating operator system
from [2].
Definition 5. Let
X
be an operator
Z
-system and
B
be a
C∗
-algebra generated by
X
. Let
π
:
B→
BZ
(
E
)be a
Z
-irreducible representation of
B
. We say that
XZ
-separates
π
if for every
Z
-irreducible
representation
σ
of
B
on self-dual Hilbert
Z
-module
F
and for every isometry
V∈BZ
(
E,F
), the
condition V∗σ(x)V=π(x)for all x∈Ximplies that σand πare unitarily equivalent representations
of B.
Proposition 4. Let
X
be an operator
Z
-system in a
C∗
-algebra
B
such that
B
=
C∗
(
X
). If
π
:
B→
BZ(E)is a Z-boundary representation of Bfor Xthen XZ-separates π.
Proof. Let
σ
:
B→BZ
(
F
)be a
Z
-irreducible representation of
B
, where
F
is self-dual Hilbert
Z
-module and
V∈BZ
(
E,F
)is an isometry such that
V∗σ
(
x
)
V
=
π
(
x
)for all
x∈X
. Since
π
is a
Z
-boundary representation for
X
and
V∗σ
(
·
)
V
is a completely positive
Z
-bimodule extension of
π|X
implies
V∗σ
(
b
)
V
=
π
(
b
)for all
b∈B
. We have
V
is isometry and [
π
(
B
)
VE
]=
F
implies
VE
is a
reducing subspace for
σ
(
B
). Also,
σ
is
Z
-irreducible implies
VE
=
F
. Thus
V
is unitary, showing that
σand πare unitarily equivalent representations. Hence XZ-separates π.
The characterization theorem for Z-boundary representations as follows:
Theorem 2. Let
X
be an operator
Z
-systems in a
C∗
-algebra
B
such that
B
=
C∗
(
X
). Let
π
be an
Z
-irreducible representation of
B
. Then
π
is a
Z
-boundary representation for
X
if and only if the
following conditions are satisfied:
(i) πis a Z-finite representation for X.
(ii) π|Xis Z-pure.
(iii) XZ-separates π.
(iv) Let K={φ∈CPZ(B,BZ(E)) : φ|X=π|X}and every φin Kis Z-extreme point of K.
https://doi.org/10.17993/3cemp.2022.110250.173-184
Proof. Suppose
π
is a
Z
-boundary representation of
B
for
X
then conditions (i), (ii),(iii) and (iv)
follows from Proposition 3, Proposition 1, Proposition 4 and Proposition 2.
Conversely, Suppose the
Z
-irreducible representation satisfies all four conditions (i), (ii), (iii), and
(iv). Let
K
=
{φ∈CPZ
(
B,BZ
(
E
)) :
φ|X
=
π|X}
. To show
π
is a
Z
-boundary representation for
X
, it is
enough to show that
K
is
{π}
. Using (iv), let
φ
be
Z
-extreme point of
K
. Now, we prove that
φ
is a
Z
-
pure in
UCPZ
(
B,BZ
(
E
)). Let
φ1,φ
2
be in
K
such that
φ1
(
b
)+
φ2
(
b
)=
φ
(
b
)for all
b∈B
. In particular,
φ1
(
x
)+
φ2
(
x
)=
φ
(
x
)for all
x∈X
. Our assumption,
φ|X
=
π|X
is pure implies there exists
ci≥
0in
Z
such that
φi
(
x
)=
ciφ
(
x
)for all
x∈X
. If
c1
=0and 1
∈X
then
φ1
(1) = 0, thus
φ1
=0. This contracts
to the choice of
φ1
, therefore
c1>
0and similarly
c2>
0. Using [16, Definition 5.1 and Proposition 5.2],
and every
Z
-extreme points are Choquet
Z
-points, we have
φi
=(
c
1
2
i
)
∗φ
(
c
1
2
i
)and (
c
1
2
1
)
∗c
1
2
1
+(
c
1
2
2
)
∗c
1
2
2
=1.
Now put
ψi
=(
c−1
2
i
)
∗φi
(
c−1
2
i
)for
i
=1
,
2. Then
ψi∈K
and (
c
1
2
1
)
∗ψ1
(
c
1
2
1
)+(
c
1
2
2
)
∗ψ2
(
c
1
2
1
)=
φ
. Since
φ
is
Z
-extreme point of
K
then there exists unitary elements
ui∈Z
for
i
=1
,
2such that
ψi
=
u∗
iφui
for
i
=1
,
2.Wehave(
c−1
2
i
)
∗φi
(
c−1
2
i
)=
u∗
iφui
for
i
=1
,
2and
c−1
iφi
=
u∗
iφui
for
i
=1
,
2. By [16, Definition
5.1 and Proposition 5.2], thus φi=ciφfor i=1,2. Hence φis a Z-pure in UCPZ(B,BZ(E)).
By Remark 2, there is a
Z
-irreducible representation
σ
of
B
on a self-dual Hilbert
Z
-module
F
and an isometry
V∈BZ
(
E,F
)such that
φ
=
V∗σV
. In particular,
π
(
x
)=
φ
(
x
)=
V∗σ
(
x
)
V
for all
x∈X
. The assumption (iii),
XZ
-separates
π
implies that
σ
is unitarily equivalent to
π
. Thus there
exists a unitary
U∈BZ
(
E,F
)such that
σ
=
U∗πU
. Therefore we have
π
(
x
)=(
UV
)
∗π
(
x
)
UV
for all
x∈X
.
UV
is isometry in
BZ
(
E
). The assumption (i),
π
is a
Z
-finite representation for
X
implies
UV
is unitary. Thus
V
=
U∗UV
is a unitary in
BZ
(
E,F
). Now
π|X
=
V∗σV |X
becomes
π
(
x
)=
V−1σ
(
x
)
V
for all
x∈X
.
V−1σV
is a representation of
B
which agrees with
π
on
X
. Therefore
π
(
b
)=
V−1σ
(
b
)
V
for all b∈B=C∗(X). Hence φ=πon B.
4 Z-HYPERRIGIDITY
In this section, we introduce the notion of
Z
-hyperrigidity in the operator
Z
-system.
Z
-hyperrigidity is
an analogue version of Arveson’s [4] notion of hyperrigidity. We define Z-hyperrigidity as follows:
Definition 6. Let
A
be a
C∗
-algebra, and let
G⊆A
(finite or countably infinite) be a set of generators
of
A
(i.e.,
A
=
C∗
(
G
)). Then
G
is said to be
Z
-hyperrigid if for every faithful representation
A⊆BZ
(
E
)
of
A
on a self-dual Hilbert
Z
-module
E
and every sequence of unital completely positive
Z
-bimodule
maps φn:BZ(E)→BZ(E),n=1,2, ...,
lim
n→∞ ||φn(g)−g|| =0,∀g∈G=⇒lim
n→∞ ||φn(a)−a|| =0,∀a∈A. (1)
We have lightened notation in the above definition by identifying the
C∗
-algebra
A
with its image
π
(
A
)in a faithful nondegenerate representation
π
:
A→BZ
(
E
)on a self-dual Hilbert
Z
-module
E
.
Notably,
Z
-hyperrigidity of operator
Z
-system on a self-dual Hilbert
Z
-module implies not only that
(1) should hold for sequences of UCP
Z
-bimodule maps
φn
defined on
BZ
(
E
), but also that the property
should hold for every other faithful representation of
A
. If
Z
=
C
, then the definition of
Z
-hyperrigity
is the same as the definition of hyperrigidity in [4, definition 1.1].
Proposition 5. Let Abe a C∗-algebra and Gbe a generating subset of A. Then Gis Z-hyperrigid if
and only if the operator Z-system generated by Gis Z-hyperrigid.
Proof. The proof follows directly from the definition of Z-hyperrigidity.
Now we prove the characterization theorem for Z-hyperrigid operator Z-systems.
Theorem 3. Let
X
be a separable operator
Z
-system and
X
generates a
C∗
-algebra
A
(i.e.,
A
=
C∗
(
X
)).
The following are equivalent:
(i) Xis Z-hyperrigid.
https://doi.org/10.17993/3cemp.2022.110250.173-184
179
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
(ii)
For every nondegenerate representation
π
:
A→BZ
(
E
)on a separable self-dual Hilbert
Z
-module
and every sequence φn:A→BZ(E)of UCP Z-bimodule maps,
lim
n→∞ ||φn(x)−π(x)|| =0∀x∈X=⇒lim
n→∞ ||φn(a)−π(a)|| =0∀a∈A.
(iii)
For every nondegenerate representation
π
:
A→BZ
(
E
)on a separable self-dual Hilbert
Z
-module,
π|Xhas the Z-unique extension property.
(iv)
For every unital
C∗
-algebra
B
, every unital homomorphism of
C∗
-algebra
θ
:
A→B
and every
UCP Z-module map φ:B→B,
φ(x)=x∀x∈θ(X)=⇒φ(x)=x∀x∈θ(A).
Proof. (i) =
⇒
(ii): Let
π
:
A→BZ
(
E
)be a nondegenerate representation on a separable self-dual Hilbert
Z
-module and let
φn
:
A→BZ
(
E
)be a sequence of UCP
Z
-module maps such that
||φn
(
x
)
−π
(
x
)
|| →
0
for all x∈X.
Let
σ
:
A→BZ
(
F
)be a faithful representation of
A
on another separable self-dual Hilbert
Z
-module
F
. Then
σ⊕π
:
A→BZ
(
F⊕E
)is a faithful representation, so that each of the
Z
-module maps
ψn:(σ⊕π)(A)→BZ(F⊕E)
ψn:σ(a)⊕π(a)→ σ(a)⊕φn(a),a∈A,
is unital completely positive
Z
-module map. By [17, Remark 4.12]
ψn
can be extended to a UCP
Z
-module map
˜
ψn
:
BZ
(
F⊕E
)
→BZ
(
F⊕E
). By our assumption
φn|X
converges to
π|X
in pointwise
norm. Thus
˜
ψn
converges in pointwise norm to the identity map on (
σ⊕π
)(
X
). Since
X
is
Z
-hyperrigid,
we have
˜
ψn
converges in pointwise norm to the identity map on (
σ⊕π
)(
A
). Therefore for every
a∈A
,
lim sup
n→∞ ||φn(a)−π(a)|| ≤ lim sup
n→∞ ||σ(a)⊕φn(a)−σ(a)⊕π(a)||
= lim
n→∞ ||˜
ψn(σ(a)⊕π(a)) −σ(a)⊕π(a)|| =0.
Hence φnconverges in pointwise norm to πon A.
(ii) =
⇒
(iii): Let
φ
:
A→BZ
(
F
)be a unital completely positive
Z
-module map such that
φ|X
=
π|X
.
Take
φn
(
X
)=
φ
(
X
)for all
n∈N
, so by hypothesis (ii),
φ
(
A
)=
π
(
A
). Thus
π|X
has the
Z
-unique
extension property.
(iii) =
⇒
(iv): Let
ρ
be a unital
∗
-homomorphism from
C∗
-algebra
A
to
C∗
-algebra
B
. Let
φ
:
B→B
be a UCP
Z
-module map.
φ
satisfies
φ
(
ρ
(
x
)) =
ρ
(
x
)
∀x∈X
. We claim that
φ
(
ρ
(
a
)) =
ρ
(
a
)
∀a∈A
.
Let B0be the separable C∗-sub algebra of Bgenerated by
ρ(A)∪φ(ρ(A)) ∪φ2(ρ(A)) ∪···.
Observe that
φ
(
B0
)
⊆B0
. Since
B0
is separable, we can faithfully represent
B0⊆BZ
(
F
)for some
separable self-dual Hilbert
Z
-module
F
. By [17, Remark 4.12], there is a UCP
Z
-module map
˜φ
:
BZ
(
F
)
→BZ
(
F
)such that
˜φ|B0
=
φ
and in particular
˜φ
(
ρ
(
x
)) =
ρ
(
x
)for
x∈X
. Since
a∈A→
ρ
(
a
)
∈BZ
(
F
)is a representation on a separable self-dual Hilbert
Z
-module. Our assumption (iii)
implies that ˜φmust fix ρ(A)elementwise. Therefore φ(ρ(a)) = ˜φ(ρ(a)) = ρ(a)∀a∈A.
(iv)=
⇒
(i): Suppose that
A⊆BZ
(
F
)is faithfully represented on some self-dual Hilbert
Z
-module
F
,
and
φ1,φ
2,···
:
BZ
(
F
)
→BZ
(
F
)is a sequence of UCP
Z
-module maps satisfying
lim
n→∞ ||φn
(
x
)
−x||
=
0∀x∈X. We claim that
lim
n→∞ ||φn(a)−a|| =0,∀a∈A.
Let
ℓ∞
(
BZ
(
F
)) denote the set of all bounded sequences with components in
BZ
(
F
)such that
ℓ∞
(
BZ
(
F
))
is a
C∗
-algebra. Let
c0
(
BZ
(
F
)) denote the set of all sequences in
ℓ∞
(
BZ
(
F
)) that converges to zero in
norm and c0(BZ(F)) is ideal in ℓ∞(BZ(F)).
https://doi.org/10.17993/3cemp.2022.110250.173-184
Define the UCP Z-module map φ0:ℓ∞(BZ(F)) →ℓ∞(BZ(F)) as follows:
φ0(a1,a
2,a
3, ...)=(φ1(a1),φ
2(a2),φ
3(a3), ...).
Thus the map
φ0
carries the ideal
c0
(
BZ
(
F
)) into itself. Hence we can define the UCP
Z
-module map
of the quotient φ:ℓ∞(BZ(F))/c0(BZ(F)) →ℓ∞(BZ(F))/c0(BZ(F)) by
φ(x+c0(BZ(F))) = φ0(x)+c0(BZ(F)),x∈ℓ∞(BZ(F)).
Now consider the natural embedding ρ:A→ℓ∞(BZ(F))/c0(BZ(F)),
ρ(a)=(a, a, a, ...)+c0(BZ(F)).
By our assumption, ||φn(x)−x|| → 0as n→∞for x∈X, and thus
φ(ρ(x)) = (φ1(x),φ
2(x), ...)+c0(BZ(F))=(x, x, ...)+c0(BZ(F)) = ρ(x).
Therefore φrestricts the identity map on ρ(X).
Applying assumption (iv) to the inclusions
ρ(X)⊆ρ(A)⊆ℓ∞(BZ(F))/c0(BZ(F))
and the UCP
Z
-module map
φ
:
ℓ∞
(
BZ
(
F
))
/c0
(
BZ
(
F
))
→ℓ∞
(
BZ
(
F
))
/c0
(
BZ
(
F
)), implies that
φ
must
fix every element of ρ(A). Since ρ(a)=(a, a, ...)+c0(BZ(F)) and
φ(ρ(a))=(φ1(a),φ
2(a), ...)+c0(BZ(F)),
hence we have (φ1(a)−a, φ2(a)−a, ...)∈c0(BZ(F)). This proves our claim.
Now we discuss the examples of
Z
-hyperrigidity. Let
V1,V
2, ..., Vn
be an arbitrary set of isometries
acting on some self-dual Hilbert
Z
-module. We exhibit a
Z
-hyperrigid generator for a
C∗
-algebra
generated by the isometries V1,V
2, ..., Vn.
Theorem 4. Let
V1,V
2, ..., Vn
be a set of isometries on some self-dual Hilbert
Z
-module and generate
aC∗-algebra A. Let
G={V1,V
2, ..., Vn,V
1V∗
1+V2V∗
2+···+VnV∗
n}
then Gis a Z-hyperrigid generator for A.
Proof. Let
X
be the operator
Z
-systems generated by
G
. By Corollary 5,
G
is hyperrigid if and only
if
X
is hyperrigid. Using Theorem 3, it is enough to prove that for every nondegenerate representation
πof A,π|Xhas the Z-unique extension property.
Consider a representation
π
:
A→BZ
(
E
)and let
W1,W
2, ..., Wn
be isometries such that
Wi
=
π(Vi),i=1,2, ..., n. Let φ:A→BZ(E)be a UCP Z-module map satisfying
φ(Vi)=Wi,1≤i≤n,
and
φ(V1V∗
1+V2V∗
2+···+VnV∗
n)=W1W∗
1+W2W∗
2+···+WnW∗
n.
Thus, φ(x)=π(x)∀x∈X. We claim that φ=πon A.
From Remark 1, Using an analogue version of the Stinespring’s dilation theorem. We can express
the UCP Z-module map φas follows:
φ(a)=W∗σ(a)W, ∀a∈A.
Where
σ
is a representation of
A
on a self-dual Hilbert
Z
-module
F
,
W
:
E→F
is an isometry, and
which is minimal in the sense that span closure of σ(A)WEis F.
https://doi.org/10.17993/3cemp.2022.110250.173-184
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
180
(ii)
For every nondegenerate representation
π
:
A→BZ
(
E
)on a separable self-dual Hilbert
Z
-module
and every sequence φn:A→BZ(E)of UCP Z-bimodule maps,
lim
n→∞ ||φn(x)−π(x)|| =0∀x∈X=⇒lim
n→∞ ||φn(a)−π(a)|| =0∀a∈A.
(iii)
For every nondegenerate representation
π
:
A→BZ
(
E
)on a separable self-dual Hilbert
Z
-module,
π|Xhas the Z-unique extension property.
(iv)
For every unital
C∗
-algebra
B
, every unital homomorphism of
C∗
-algebra
θ
:
A→B
and every
UCP Z-module map φ:B→B,
φ(x)=x∀x∈θ(X)=⇒φ(x)=x∀x∈θ(A).
Proof. (i) =
⇒
(ii): Let
π
:
A→BZ
(
E
)be a nondegenerate representation on a separable self-dual Hilbert
Z
-module and let
φn
:
A→BZ
(
E
)be a sequence of UCP
Z
-module maps such that
||φn
(
x
)
−π
(
x
)
|| →
0
for all x∈X.
Let
σ
:
A→BZ
(
F
)be a faithful representation of
A
on another separable self-dual Hilbert
Z
-module
F
. Then
σ⊕π
:
A→BZ
(
F⊕E
)is a faithful representation, so that each of the
Z
-module maps
ψn:(σ⊕π)(A)→BZ(F⊕E)
ψn:σ(a)⊕π(a)→ σ(a)⊕φn(a),a∈A,
is unital completely positive
Z
-module map. By [17, Remark 4.12]
ψn
can be extended to a UCP
Z
-module map
˜
ψn
:
BZ
(
F⊕E
)
→BZ
(
F⊕E
). By our assumption
φn|X
converges to
π|X
in pointwise
norm. Thus
˜
ψn
converges in pointwise norm to the identity map on (
σ⊕π
)(
X
). Since
X
is
Z
-hyperrigid,
we have
˜
ψn
converges in pointwise norm to the identity map on (
σ⊕π
)(
A
). Therefore for every
a∈A
,
lim sup
n→∞ ||φn(a)−π(a)|| ≤ lim sup
n→∞ ||σ(a)⊕φn(a)−σ(a)⊕π(a)||
= lim
n→∞ ||˜
ψn(σ(a)⊕π(a)) −σ(a)⊕π(a)|| =0.
Hence φnconverges in pointwise norm to πon A.
(ii) =
⇒
(iii): Let
φ
:
A→BZ
(
F
)be a unital completely positive
Z
-module map such that
φ|X
=
π|X
.
Take
φn
(
X
)=
φ
(
X
)for all
n∈N
, so by hypothesis (ii),
φ
(
A
)=
π
(
A
). Thus
π|X
has the
Z
-unique
extension property.
(iii) =
⇒
(iv): Let
ρ
be a unital
∗
-homomorphism from
C∗
-algebra
A
to
C∗
-algebra
B
. Let
φ
:
B→B
be a UCP
Z
-module map.
φ
satisfies
φ
(
ρ
(
x
)) =
ρ
(
x
)
∀x∈X
. We claim that
φ
(
ρ
(
a
)) =
ρ
(
a
)
∀a∈A
.
Let B0be the separable C∗-sub algebra of Bgenerated by
ρ(A)∪φ(ρ(A)) ∪φ2(ρ(A)) ∪···.
Observe that
φ
(
B0
)
⊆B0
. Since
B0
is separable, we can faithfully represent
B0⊆BZ
(
F
)for some
separable self-dual Hilbert
Z
-module
F
. By [17, Remark 4.12], there is a UCP
Z
-module map
˜φ
:
BZ
(
F
)
→BZ
(
F
)such that
˜φ|B0
=
φ
and in particular
˜φ
(
ρ
(
x
)) =
ρ
(
x
)for
x∈X
. Since
a∈A→
ρ
(
a
)
∈BZ
(
F
)is a representation on a separable self-dual Hilbert
Z
-module. Our assumption (iii)
implies that ˜φmust fix ρ(A)elementwise. Therefore φ(ρ(a)) = ˜φ(ρ(a)) = ρ(a)∀a∈A.
(iv)=
⇒
(i): Suppose that
A⊆BZ
(
F
)is faithfully represented on some self-dual Hilbert
Z
-module
F
,
and
φ1,φ
2,···
:
BZ
(
F
)
→BZ
(
F
)is a sequence of UCP
Z
-module maps satisfying
lim
n→∞ ||φn
(
x
)
−x||
=
0∀x∈X. We claim that
lim
n→∞ ||φn(a)−a|| =0,∀a∈A.
Let
ℓ∞
(
BZ
(
F
)) denote the set of all bounded sequences with components in
BZ
(
F
)such that
ℓ∞
(
BZ
(
F
))
is a
C∗
-algebra. Let
c0
(
BZ
(
F
)) denote the set of all sequences in
ℓ∞
(
BZ
(
F
)) that converges to zero in
norm and c0(BZ(F)) is ideal in ℓ∞(BZ(F)).
https://doi.org/10.17993/3cemp.2022.110250.173-184
Define the UCP Z-module map φ0:ℓ∞(BZ(F)) →ℓ∞(BZ(F)) as follows:
φ0(a1,a
2,a
3, ...)=(φ1(a1),φ
2(a2),φ
3(a3), ...).
Thus the map
φ0
carries the ideal
c0
(
BZ
(
F
)) into itself. Hence we can define the UCP
Z
-module map
of the quotient φ:ℓ∞(BZ(F))/c0(BZ(F)) →ℓ∞(BZ(F))/c0(BZ(F)) by
φ(x+c0(BZ(F))) = φ0(x)+c0(BZ(F)),x∈ℓ∞(BZ(F)).
Now consider the natural embedding ρ:A→ℓ∞(BZ(F))/c0(BZ(F)),
ρ(a)=(a, a, a, ...)+c0(BZ(F)).
By our assumption, ||φn(x)−x|| → 0as n→∞for x∈X, and thus
φ(ρ(x)) = (φ1(x),φ
2(x), ...)+c0(BZ(F))=(x, x, ...)+c0(BZ(F)) = ρ(x).
Therefore φrestricts the identity map on ρ(X).
Applying assumption (iv) to the inclusions
ρ(X)⊆ρ(A)⊆ℓ∞(BZ(F))/c0(BZ(F))
and the UCP
Z
-module map
φ
:
ℓ∞
(
BZ
(
F
))
/c0
(
BZ
(
F
))
→ℓ∞
(
BZ
(
F
))
/c0
(
BZ
(
F
)), implies that
φ
must
fix every element of ρ(A). Since ρ(a)=(a, a, ...)+c0(BZ(F)) and
φ(ρ(a))=(φ1(a),φ
2(a), ...)+c0(BZ(F)),
hence we have (φ1(a)−a, φ2(a)−a, ...)∈c0(BZ(F)). This proves our claim.
Now we discuss the examples of
Z
-hyperrigidity. Let
V1,V
2, ..., Vn
be an arbitrary set of isometries
acting on some self-dual Hilbert
Z
-module. We exhibit a
Z
-hyperrigid generator for a
C∗
-algebra
generated by the isometries V1,V
2, ..., Vn.
Theorem 4. Let
V1,V
2, ..., Vn
be a set of isometries on some self-dual Hilbert
Z
-module and generate
aC∗-algebra A. Let
G={V1,V
2, ..., Vn,V
1V∗
1+V2V∗
2+···+VnV∗
n}
then Gis a Z-hyperrigid generator for A.
Proof. Let
X
be the operator
Z
-systems generated by
G
. By Corollary 5,
G
is hyperrigid if and only
if
X
is hyperrigid. Using Theorem 3, it is enough to prove that for every nondegenerate representation
πof A,π|Xhas the Z-unique extension property.
Consider a representation
π
:
A→BZ
(
E
)and let
W1,W
2, ..., Wn
be isometries such that
Wi
=
π(Vi),i=1,2, ..., n. Let φ:A→BZ(E)be a UCP Z-module map satisfying
φ(Vi)=Wi,1≤i≤n,
and
φ(V1V∗
1+V2V∗
2+···+VnV∗
n)=W1W∗
1+W2W∗
2+···+WnW∗
n.
Thus, φ(x)=π(x)∀x∈X. We claim that φ=πon A.
From Remark 1, Using an analogue version of the Stinespring’s dilation theorem. We can express
the UCP Z-module map φas follows:
φ(a)=W∗σ(a)W, ∀a∈A.
Where
σ
is a representation of
A
on a self-dual Hilbert
Z
-module
F
,
W
:
E→F
is an isometry, and
which is minimal in the sense that span closure of σ(A)WEis F.
https://doi.org/10.17993/3cemp.2022.110250.173-184
181
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
First we prove that σ(Vi)W=WW
i,1≤i≤n. For i=1,2, ...n we have
W∗σ(Vi)WW∗σ(Vi)V=φ(Vi)∗φ(Vi)=W∗
iWi=1E,
thus
W∗σ
(
Vi
)(
1−WW∗
)
σ
(
Vi
)
W
=0, therefore
WE
is invariant under
σ
(
Vi
). Hence we get
σ
(
Vi
)
W
=
WW∗σ(Vi)W=Wφ(Vi)=WW
i.
Next, since n
i=1
WiW∗
i=π(
n
i=1
ViV∗
i)=φ(
n
i=1
ViV∗
i), we get
n
i=1
σ(Vi)WW∗σ(Vi)∗=
n
i=1
WW
iW∗
iW∗
=Wφ(
n
i=1
ViV∗
i)W∗
=WW∗n
i=1
σ(ViV∗
i)WW∗
=
n
i=1
WW∗σ(Vi)σ(V∗
i)WW∗.
We know that
σ
(
Vi
)
W
=
WW∗σ
(
Vi
)
W
for all
i
. In the above equations, subtract the left side from the
right, and we have n
i=1
WW∗σ(Vi)(1F−WW∗)σ(Vi)∗WW∗=0.
Thus (
1F−WW∗
)
σ
(
Vi
)
∗WW∗
=0for all
i
=1
,
2
, ..., n
. Therefore
WE
is invariant under both
σ
(
Vi
)
and
σ
(
Vi
)
∗
for all
i
=1
,
2
, ..., n
. Since the
C∗
-algebra
A
is generated by the
Vi
, we have
σ
(
A
)
WE⊆WE
.
By the minimality condition, we have
WE
=
F
. Thus
W
is unitary. Therefore
φ
(
a
)=
W−1σ
(
a
)
W
is a representation on
A
. By our assumption,
φ
agrees with
π
on a generating set. Hence
φ
=
π
on
C∗-algebra A.
The Cuntz algebras
On
is the universal
C∗
-algebra generated by isometries
V1,V
2, ..., Vn
such that
V1V∗
1
+
V2V∗
2
+
···
+
VnV∗
n
=
1
. We can discard the identity operator from the generating set
G
to
conclude the above result.
Corollary 1. The set G={V1,V
2, ..., Vn}of generators of the Cuntz algebra Onis Z-hyperrigid.
Theorem 5. Let
X
be a separable operator
Z
-system generating a
C∗
-algebra
A
. If
X
is
Z
-hyperrigid
then every Z-irreducible representation of Ais a Z-boundary representation for X.
Proof. Suppose
X
is an operator
Z
-system in a
C∗
-algebra
A
. Then by Theorem 3, every nondegene-
rate representation of
A
on separable self-dual Hilbert
Z
-module has the
Z
-unique extension property
when nondegenerate representation restricted to
X
. Since every
Z
-irreducible representation of a
C∗
-algebra
A
is a nondegenerate representation of
A
. Therefore, every
Z
-irreducible representation of
A
on separable self-dual Hilbert
Z
-module has the
Z
-unique extension property when
Z
-irreducible repre-
sentation restricted to
X
. Hence, every
Z
-irreducible representation of
A
is a
Z
-boundary representation
for X.
Problem 1. Let
X
be a separable operator
Z
-system generating a
C∗
-algebra
A
. If every
Z
-irreducible
representation of
C∗
-algebra
A
is a
Z
-boundary representation for a separable operator
Z
-system
X⊆A
.
Then Xis Z-hyperrigid.
Proposition 6. Let
X
be an operator
Z
-system generating a
C∗
-algebra
A
=
C∗
(
X
). Let
πi
:
A→B
(
Ei
)
be a representation on a self-dual Hilbert
Z
-module such that
πi|X
has the
Z
-unique extension property
for each iin an index set I. Then the direct sum of UCP Z-module maps
π=⊕i∈Iπi|X:X→B(⊕i∈IEi)
has the Z-unique extension property.
https://doi.org/10.17993/3cemp.2022.110250.173-184
Proof. Let
φ
:
A→B
(
⊕i∈IEi
)be a UCP
Z
-module map such that
π|X
=
φ|X
. For each
i∈I
, let
φi:A→B(Ei)be the UCP Z-module map such that
φi(a)=Piφ(a)|Ei,a∈A
where
Pi
is the projection from
⊕i∈IEi
onto
Ei
. Observe that
φ|X
=
π|X
. Our assumption
πi|X
has
Z
-unique extension property implies that
φi
(
a
)=
πi
(
a
)for all
a∈A
. Equivalently, we have
Piφ(a)Pi=π(a)Pifor all a∈A. Using the Schwarz inequality of φ, we have
Piφ(a)∗(1−Pi)φ(a)Pi=Piφ(a)∗φ(a)Pi−Piφ(a)∗Piφ(a)Pi
≤Piφ(a∗a)Pi−Piφ(a)∗Piφ(a)Pi
=π(a∗a)Pi−π(a)∗π(a)Pi=0.
Therefore,
|
(
1−Pi
)
φ
(
a
)
Pi|2
=0. Thus it follows that
Pi
commutes with the self-adjoint family of
operators φ(A). Hence for every a∈A, we have
φ(a)=
i∈I
φ(a)Pi=
i∈I
Piφ(a)Pi=
i∈I
π(a)Pi=π(a).
Let
A
be a separable
C∗
-algebra. The set of unitary equivalence classes of
Z
-irreducible representa-
tions of Ais said to be a spectrum of A.
Theorem 6. Let
X
be a separable operator
Z
-system generating a
C∗
-algebra
A
and let
A
have a
countable spectrum. If every
Z
-irreducible representation of
A
is a
Z
-boundary representation for
X
then Xis Z-hyperrigid.
Proof. By the Theorem 3, it is enough to prove that for every representation
π
:
A→B
(
E
)of
A
on a separable self-dual Hilbert
Z
-module, the UCP
Z
-module map
π|X
has the
Z
-unique extension
property. Since the spectrum
A
is countable,
A
is the type I
C∗
-algebra. Therefore
π
can be decomposed
uniquely into a direct integral of mutually disjoint type I factor representation. Because the spectrum
A
is countable, the direct integral must be a countable direct sum. Hence
π
can be decomposed into a
direct sum of subrepresentations
πn
:
A→B
(
En
)of
A
on a separable self-dual Hilbert
Z
-modules. Thus
E=E1⊕E
2⊕··· π=π1⊕π2⊕···
With the property that each
πn
is unitarily equivalent to a finite or countable direct sum of copies of
a single
Z
-irreducible representations
σn
:
A→B
(
Fn
)of
A
on a separable self-dual Hilbert
Z
-modules.
By our assumption, each UCP
Z
-module map
σn|X
has the
Z
-unique extension property. Therefore
the above decomposition of
π|X
can be expressed as a double direct sum of UCP
Z
-module maps
with the
Z
-unique extension property. Using Proposition 6, we have
π|X
has the
Z
-unique extension
property.
ACKNOWLEDGMENT
The first author thanks the Council of Scientific & Industrial Research (CSIR) for providing a doctoral
fellowship. The second 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.,Shankar, P., and Vijayarajan, A. K. (2021). Boundary representations
and rectangular hyperrigidity. Banach J. Math. Anal., 15, no. 2, Paper No. 38, 19 pp.
https://doi.org/10.17993/3cemp.2022.110250.173-184
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
182
First we prove that σ(Vi)W=WW
i,1≤i≤n. For i=1,2, ...n we have
W∗σ(Vi)WW∗σ(Vi)V=φ(Vi)∗φ(Vi)=W∗
iWi=1E,
thus
W∗σ
(
Vi
)(
1−WW∗
)
σ
(
Vi
)
W
=0, therefore
WE
is invariant under
σ
(
Vi
). Hence we get
σ
(
Vi
)
W
=
WW∗σ(Vi)W=Wφ(Vi)=WW
i.
Next, since n
i=1
WiW∗
i=π(
n
i=1
ViV∗
i)=φ(
n
i=1
ViV∗
i), we get
n
i=1
σ(Vi)WW∗σ(Vi)∗=
n
i=1
WW
iW∗
iW∗
=Wφ(
n
i=1
ViV∗
i)W∗
=WW∗n
i=1
σ(ViV∗
i)WW∗
=
n
i=1
WW∗σ(Vi)σ(V∗
i)WW∗.
We know that
σ
(
Vi
)
W
=
WW∗σ
(
Vi
)
W
for all
i
. In the above equations, subtract the left side from the
right, and we have n
i=1
WW∗σ(Vi)(1F−WW∗)σ(Vi)∗WW∗=0.
Thus (
1F−WW∗
)
σ
(
Vi
)
∗WW∗
=0for all
i
=1
,
2
, ..., n
. Therefore
WE
is invariant under both
σ
(
Vi
)
and
σ
(
Vi
)
∗
for all
i
=1
,
2
, ..., n
. Since the
C∗
-algebra
A
is generated by the
Vi
, we have
σ
(
A
)
WE⊆WE
.
By the minimality condition, we have
WE
=
F
. Thus
W
is unitary. Therefore
φ
(
a
)=
W−1σ
(
a
)
W
is a representation on
A
. By our assumption,
φ
agrees with
π
on a generating set. Hence
φ
=
π
on
C∗-algebra A.
The Cuntz algebras
On
is the universal
C∗
-algebra generated by isometries
V1,V
2, ..., Vn
such that
V1V∗
1
+
V2V∗
2
+
···
+
VnV∗
n
=
1
. We can discard the identity operator from the generating set
G
to
conclude the above result.
Corollary 1. The set G={V1,V
2, ..., Vn}of generators of the Cuntz algebra Onis Z-hyperrigid.
Theorem 5. Let
X
be a separable operator
Z
-system generating a
C∗
-algebra
A
. If
X
is
Z
-hyperrigid
then every Z-irreducible representation of Ais a Z-boundary representation for X.
Proof. Suppose
X
is an operator
Z
-system in a
C∗
-algebra
A
. Then by Theorem 3, every nondegene-
rate representation of
A
on separable self-dual Hilbert
Z
-module has the
Z
-unique extension property
when nondegenerate representation restricted to
X
. Since every
Z
-irreducible representation of a
C∗
-algebra
A
is a nondegenerate representation of
A
. Therefore, every
Z
-irreducible representation of
A
on separable self-dual Hilbert
Z
-module has the
Z
-unique extension property when
Z
-irreducible repre-
sentation restricted to
X
. Hence, every
Z
-irreducible representation of
A
is a
Z
-boundary representation
for X.
Problem 1. Let
X
be a separable operator
Z
-system generating a
C∗
-algebra
A
. If every
Z
-irreducible
representation of
C∗
-algebra
A
is a
Z
-boundary representation for a separable operator
Z
-system
X⊆A
.
Then Xis Z-hyperrigid.
Proposition 6. Let
X
be an operator
Z
-system generating a
C∗
-algebra
A
=
C∗
(
X
). Let
πi
:
A→B
(
Ei
)
be a representation on a self-dual Hilbert
Z
-module such that
πi|X
has the
Z
-unique extension property
for each iin an index set I. Then the direct sum of UCP Z-module maps
π=⊕i∈Iπi|X:X→B(⊕i∈IEi)
has the Z-unique extension property.
https://doi.org/10.17993/3cemp.2022.110250.173-184
Proof. Let
φ
:
A→B
(
⊕i∈IEi
)be a UCP
Z
-module map such that
π|X
=
φ|X
. For each
i∈I
, let
φi:A→B(Ei)be the UCP Z-module map such that
φi(a)=Piφ(a)|Ei,a∈A
where
Pi
is the projection from
⊕i∈IEi
onto
Ei
. Observe that
φ|X
=
π|X
. Our assumption
πi|X
has
Z
-unique extension property implies that
φi
(
a
)=
πi
(
a
)for all
a∈A
. Equivalently, we have
Piφ(a)Pi=π(a)Pifor all a∈A. Using the Schwarz inequality of φ, we have
Piφ(a)∗(1−Pi)φ(a)Pi=Piφ(a)∗φ(a)Pi−Piφ(a)∗Piφ(a)Pi
≤Piφ(a∗a)Pi−Piφ(a)∗Piφ(a)Pi
=π(a∗a)Pi−π(a)∗π(a)Pi=0.
Therefore,
|
(
1−Pi
)
φ
(
a
)
Pi|2
=0. Thus it follows that
Pi
commutes with the self-adjoint family of
operators φ(A). Hence for every a∈A, we have
φ(a)=
i∈I
φ(a)Pi=
i∈I
Piφ(a)Pi=
i∈I
π(a)Pi=π(a).
Let
A
be a separable
C∗
-algebra. The set of unitary equivalence classes of
Z
-irreducible representa-
tions of Ais said to be a spectrum of A.
Theorem 6. Let
X
be a separable operator
Z
-system generating a
C∗
-algebra
A
and let
A
have a
countable spectrum. If every
Z
-irreducible representation of
A
is a
Z
-boundary representation for
X
then Xis Z-hyperrigid.
Proof. By the Theorem 3, it is enough to prove that for every representation
π
:
A→B
(
E
)of
A
on a separable self-dual Hilbert
Z
-module, the UCP
Z
-module map
π|X
has the
Z
-unique extension
property. Since the spectrum
A
is countable,
A
is the type I
C∗
-algebra. Therefore
π
can be decomposed
uniquely into a direct integral of mutually disjoint type I factor representation. Because the spectrum
A
is countable, the direct integral must be a countable direct sum. Hence
π
can be decomposed into a
direct sum of subrepresentations
πn
:
A→B
(
En
)of
A
on a separable self-dual Hilbert
Z
-modules. Thus
E=E1⊕E
2⊕··· π=π1⊕π2⊕···
With the property that each
πn
is unitarily equivalent to a finite or countable direct sum of copies of
a single
Z
-irreducible representations
σn
:
A→B
(
Fn
)of
A
on a separable self-dual Hilbert
Z
-modules.
By our assumption, each UCP
Z
-module map
σn|X
has the
Z
-unique extension property. Therefore
the above decomposition of
π|X
can be expressed as a double direct sum of UCP
Z
-module maps
with the
Z
-unique extension property. Using Proposition 6, we have
π|X
has the
Z
-unique extension
property.
ACKNOWLEDGMENT
The first author thanks the Council of Scientific & Industrial Research (CSIR) for providing a doctoral
fellowship. The second 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).
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