Matrix Ordered Operator Algebras
7 Matrix Ordered Operator Algebras.
Ekaterina Juschenko, Stanislav Popovych
Department of Mathematics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
jushenko@math.chalmers.se
stanislp@math.chalmers.se
Abstract
We study the question when for a given ∗-algebra A a sequence of
cones Cn ∈ Mn(A) can be realized as cones of positive operators in a
faithful ∗-representation of A on a Hilbert space. A characterization
of operator algebras which are completely boundedly isomorphic to
∗-algebras is presented.
KEYWORDS: ∗-algebra, operator algebra, C∗-algebra, completely
bounded homomorphism, Kadison’s problem.
1 Introduction
Effros and Choi gave in [2] an abstract characterization of the self-adjoint sub-
spaces S in C∗-algebras with hierarchy of cones of positive elements inMn(S).
In Section 2 of the present paper we are concerned with the same question
for ∗-subalgebras of C∗-algebras. More precisely, let A be an associative
∗-algebra with unit. In Theorem 2 we present a characterization of the col-
lections of cones Cn ⊆Mn(A) for which there exist faithful ∗-representation
π of A on a Hilbert space H such that Cn coincides with the cone of positive
operators contained in π(n)(Mn(A)). Here π
(n)((xi,j)) = (π(xi,j)) for every
matrix (xi,j) ∈Mn(A). Note that we do not assume that A has any faithful
0 2000 Mathematics Subject Classification: 46L05, 46L07 (Primary) 47L55, 47L07,
47L30 (Secondary)
http://arxiv.org/abs/0704.1553v2
∗-representation. This follows from the requirements imposed on the cones.
In terms close to Effros and Choi we give an abstract characterizations of
matrix ordered (not necessary closed) operator ∗-algebras up to complete
order ∗-isomorphism.
Based on this characterization we study the question when an operator
algebra is similar to a C∗-algebra.
Let B be a unital (closed) operator algebra in B(H). In [8] C. Le Merdy
presented necessary and sufficient conditions for B to be self-adjoint. These
conditions involve all completely isometric representations of B on Hilbert
spaces. Our characterization is different in the following respect. If S is a
bounded invertible operator in B(H) and A is a C∗-algebra in B(H) then the
operator algebra S−1AS is not necessarily self-adjoint but only isomorphic to
a C∗-algebra via completely bounded isomorphism with completely bounded
inverse. By Haagerup’s theorem every completely bounded isomorphism π
from a C∗-algebra A to an operator algebra B has the form π(a) = S−1ρ(a)S,
a ∈ A, for some ∗-isomorphism ρ : A → B(H) and invertible S ∈ B(H).
Thus the question whether an operator algebra B is completely boundedly
isomorphic to a C∗-algebra via isomorphism which has completely bounded
inverse, is equivalent to the question if there is bounded invertible operator
S such that SBS−1 is a C∗-algebra.
We will present a criterion for an operator algebra B to be completely
boundedly isomorphic to a C∗-algebra in terms of the existence of a collection
of cones Cn ∈ Mn(B) satisfying certain axioms (see def. 3). The axioms are
derived from the properties of the cones of positive elements of a C∗-algebra
preserved under completely bounded isomorphisms.
The main results are contained in section 2. We define a ∗-admissible
sequence of cones in an operator algebra and present a criterion in Theorem 4
for an operator algebra to be completely boundedly isomorphic to a C∗-
algebra.
In the last section we consider the operator algebras and collections of
cones associated with Kadison similarity problem.
2 Operator realizations of matrix-ordered ∗-
algebras.
The aim of this section is to give necessary and sufficient conditions on a
sequences of cones Cn ⊆ Mn(A)sa for a unital ∗-algebra A such that Cn
coincides with the coneMn(A)∩Mn(B(H))
+ for some realization of A as a ∗-
subalgebra of B(H), where Mn(B(H))
+ denotes the set of positive operators
acting on Hn = H ⊕ . . .⊕H .
In [11] it was proved that a ∗-algebra A with unit e is a ∗-subalgebra of
B(H) if and only if there is an algebraically admissible cone on A such that
e is an Archimedean order unit. Applying this result to some inductive limit
of M2n(A) we obtain the desired characterization in Theorem 2.
First we give necessary definitions and fix notations. Let Asa denote the
set of self-adjoint elements in A. A subset C ⊂ Asa containing unit e of A
is algebraically admissible cone (see [12]) provided that
(i) C is a cone in Asa, i.e. λx+ βy ∈ C for all x, y ∈ C and λ ≥ 0, β ≥ 0,
λ, β ∈ R;
(ii) C ∩ (−C) = {0};
(iii) xCx∗ ⊆ C for every x ∈ A;
We call e ∈ Asa an order unit if for every x ∈ Asa there exists r > 0 such
that re+ x ∈ C. An order unit e is Archimedean if re+ x ∈ C for all r > 0
implies that x ∈ C
In what follows we will need the following.
Theorem 1. Let A be a ∗-algebra with unit e and C ⊆ Asa be a cone
containing e. If xCx∗ ⊆ C for every x ∈ A and e is an Archimedean
order unit then there is a unital ∗-representation π : A → B(H) such that
π(C) = π(Asa) ∩ B(H)
+. Moreover
1. ‖π(x)‖ = inf{r > 0 : r2 ± x∗x ∈ C}.
2. ker π = {x : x∗x ∈ C ∩ (−C)}.
3. If C ∩ (−C) = {0} then ker π = {0}, ‖π(a)‖ = inf{r > 0 : r ± a ∈ C}
for all a = a∗ ∈ A and π(C) = π(A) ∩B(H)+
Proof. Following the same lines as in [11] one obtains that the function ‖ · ‖ :
Asa → R+ defined as
‖a‖ = inf{r > 0 : re± a ∈ C}
is a seminorm on R-space Asa and |x| =
‖x∗x‖ for x ∈ A defines a pre-
C∗-norm on A. If N denote the null-space of | · | then the completion B =
A/N with respect to this norm is a C∗-algebra and canonical epimorphism
π : A → A/N extends to a unital ∗-homomorphism π : A → B. We can
assume without loss of generality that B is a concrete C∗-algebra in B(H)
for some Hilbert space H . Thus π : A → B(H) can be regarded as a unital
∗-representation. Clearly,
‖π(x)‖ = |x| for all x ∈ A.
This implies 1.
To show 2 take x ∈ ker π then ‖π(x)‖ = 0 and re ± x∗x ∈ C for all
r > 0. Since e is an Archimedean unit we have x∗x ∈ C ∩ (−C). Conversely
if x∗x ∈ C ∩ (−C) then re± x∗x ∈ C, for all r > 0, hence ‖π(x)‖ = 0 and 2
holds.
Let us prove that π(C) = π(Asa) ∩ B(H)
+. Let x ∈ Asa and π(x) ≥ 0.
Then there exists a constant λ > 0 such that ‖λIH − π(x)‖ ≤ λ, hence
|λe − x| ≤ λ. Since ‖a‖ ≤ |a| for all self-adjoint a ∈ A, see Lemma 3.3 of
[11], we have ‖λe−x‖ ≤ λ. Thus given ε > 0 we have (λ+ε)e±(λe−x) ∈ C.
Hence εe+ x ∈ C. Since e is Archimedean x ∈ C.
Conversely, let x ∈ C. To show that π(x) ≥ 0 it is sufficient to find λ > 0
such that ‖λIH − π(x)‖ ≤ λ. Since ‖λIH − π(x)‖ = |λe − x| we will prove
that |λe − x| ≤ λ for some λ > 0. From the definition of norm | · | we have
the following equivalences:
|λe− x| ≤ λ ⇔ (λ+ ε)2e− (λe− x)2 ∈ C for all ε > 0 (1)
⇔ ε1e + x(2λe− x) ≥ 0, for all ε1 > 0. (2)
By condition (iii) in the definition of algebraically admissible cone we
have that xyx ∈ C and yxy ∈ C for every x, y ∈ C. If xy = yx then
xy(x + y) ∈ C. Since e is an order unit we can choose r > 0 such that
re − x ∈ C. Put y = re − x to obtain rx(re − x) ∈ C. Hence (2) is
satisfied with λ = r
. Thus ‖λe − π(x)‖ ≤ λ and π(x) ≥ 0, which proves
π(C) = π(Asa) ∩ B(H)
In particular, for a = a∗ we have
‖π(a)‖ = inf{r > 0 : rIH ± π(a) ∈ π(C)}. (3)
We now in a position to prove 3. Suppose that C∩ (−C) = 0. Then ker π
is a ∗-ideal and ker π 6= 0 implies that there exists a self-adjoint 0 6= a ∈ ker π,
i.e. |a| = 0. Inequality ‖a‖ ≤ |a| implies re± a ∈ C for all r > 0. Since e is
Archimedean, ±a ∈ C, i.e. a ∈ C ∩ (−C) and, consequently, a = 0.
Since ker π = 0 the inclusion rIH±π(a) ∈ π(C) is equivalent to re±a ∈ C,
and by (3), ‖π(a)‖ = inf{r > 0 : re± a ∈ C}. Moreover if π(a) = π(a)∗ then
a = a∗. Thus we have π(C) = π(A) ∩ B(H)+.
We say that a ∗-algebra A with unit e is a matrix ordered if the following
conditions hold:
(a) for each n ≥ 1 we are given a cone Cn in Mn(A)sa and e ∈ C1,
(b) Cn ∩ (−Cn) = {0} for all n,
(c) for all n and m and all A ∈Mn×m(A), we have that A
∗CnA ⊆ Cm,
We call e ∈ Asa a matrix order unit provided that for every n ∈ N
and every x ∈ Mn(A)sa there exists r > 0 such that ren + x ∈ Cn, where
en = e ⊗ In. A matrix order unit is called Archimedean matrix order unit
provided that for all n ∈ N inclusion ren + x ∈ Cn for all r > 0 implies that
x ∈ Cn.
Let π : A → B(H) be a ∗-representation. Define π(n) : Mn(A) →
Mn(B(H)) by π
(n)((aij)) = (π(aij)).
Theorem 2. If A is a matrix-ordered ∗-algebra with a unit e which is
Archimedean matrix order unit then there exists a Hilbert spaceH and a faith-
ful unital ∗-representation τ : A → B(H), such that τ (n)(Cn) = Mn(τ(A))
for all n. Conversely, every unital ∗-subalgebra D of B(H) is matrix-ordered
by cones Mn(D)
+ = Mn(D) ∩ B(H)
+ and the unit of this algebra is an
Archimedean order unit.
Proof. Consider an inductive system of ∗-algebras and unital injective ∗-
homomorphisms:
φn :M2n(A) →M2n+1(A), φn(a) =
for all n ≥ 0, a ∈M2n(A).
Let B = lim
M2n(A) be the inductive limit of this system. By (c) in the
definition of the matrix ordered algebra we have φn(C2n) ⊆ C2n+1. We will
identify M2n(A) with a subalgebra of B via canonical inclusions. Let C =⋃
C2n ⊆ Bsa and let e∞ be the unit of B.
Let us prove that C is an algebraically admissible cone. Clearly, C satisfies
conditions (i) and (ii) of definition of algebraically admissible cone. To prove
(iii) suppose that x ∈ B and a ∈ C, then for sufficiently large n we have
a ∈ C2n and x ∈ M2n(A). Therefore, by (c), x
∗ax ∈ C. Thus (iii) is proved.
Since e is an Archimedean matrix order unit we obviously have that e∞ is
also an Archimedean order unit. Thus ∗-algebra B satisfies assumptions of
Theorem 1 and there is a faithful ∗-representation π : B → B(H) such that
π(C) = π(B) ∩ B(H)+.
Let ξn : M2n(A) → B be canonical injections (n ≥ 0). Then τ = π ◦ ξ0 :
A → B(H) is an injective ∗-homomorphism.
We claim that τ (2
n) is unitary equivalent to π◦ξn. By replacing π with π
where α is an infinite cardinal, we can assume that πα is unitary equivalent
to π. Since π◦ξn :M2n(A) → B(H) is a ∗-homomorphism there exist unique
Hilbert space Kn, ∗-homomorphism ρn : A → B(Kn) and unitary operator
Un : Kn ⊗ C
2n → H such that
π ◦ ξn = Un(ρn ⊗ idM2n )U
For a ∈ A, we have
π ◦ ξ0(a) = π ◦ ξn(a⊗ E2n)
= Un(ρn(a)⊗ E2n)U
where E2n is the identity matrix in M2n(C). Thus τ(a) = U0ρ0(a)U
Un(ρn(a) ⊗ E2n)U
n. Let ∼ stands for the unitary equivalence of represen-
tations. Since π ◦ ξn ∼ ρn ⊗ idM2n and π
α ∼ π we have that ραn ⊗ idM2n ∼
πα ◦ ξn ∼ ρn ⊗ idM2n . Hence ρ
n ∼ ρn. Thus ρn ⊗ E2n ∼ ρ
n ∼ ρn.
Consequently ρ0 ∼ ρn and π ◦ ξn ∼ ρ0 ⊗ idM2n ∼ τ ⊗ idM2n . Therefore
n) = τ ⊗ idM2n is unitary equivalent to π ◦ ξn.
What is left to show is that τ (n)(Cn) = Mn(τ(A))
+. Note that π ◦
ξn(M2n(A))∩B(H)
+ = π(C2n). Indeed, the inclusion π ◦ ξ(C2n) ⊆M2n(A)∩
B(H)+ is obvious. To show the converse take x ∈M2n(A) such that π(x) ≥ 0.
Then x ∈ C∩M2n(A). Using (c) one can easily show that C∩M2n(A) = C2n .
Hence π ◦ ξn(M2n(A)) ∩ B(H)
+ = π(C2n). Since τ
(2n) is unitary equivalent
to π ◦ ξn we have that τ
(2n)(C2n) =M2n(τ(A)) ∩ B(H
2n)+.
Let now show that τ (n)(Cn) =Mn(τ(A))
+. For X ∈Mn(A) denote
X 0n×(2n−n)
0(2n−n)×n 0(2n−n)×(2n−n)
∈M2n(A).
Then, clearly, τ (n)(X) ≥ 0 if and only if τ (2
n)(X̃) ≥ 0. Thus τ (n)(X) ≥ 0 is
equivalent to X̃ ∈ C2n which in turn is equivalent to X ∈ Cn by (c).
3 Operator Algebras completely boundedly
isomorphic to C∗-algebras.
The algebra Mn(B(H)) of n× n matrices with entries in B(H) has a norm
‖ · ‖n via the identification ofMn(B(H)) with B(H
n), where Hn is the direct
sum of n copies of a Hilbert space H . If A is a subalgebra of B(H) then
Mn(A) inherits a norm ‖·‖n via natural inclusion intoMn(B(H)). The norms
‖ · ‖n are called matrix norms on the operator algebra A. In the sequel all
operator algebras will be assumed to be norm closed.
Operator algebras A and B are called completely boundedly isomorphic
if there is a completely bounded isomorphism τ : A → B with completely
bounded inverse. The aim of this section is to give necessary and sufficient
conditions for an operator algebra to be completely boundedly isomorphic
to a C∗-algebra. To do this we introduce a concept of ∗-admissible cones
which reflect the properties of the cones of positive elements of a C∗-algebra
preserved under completely bounded isomorphism.
Definition 3. Let B be an operator algebra with unit e. A sequence Cn ⊆
Mn(B) of closed (in the norm ‖ · ‖n) cones will be called ∗-admissible if it
satisfies the following conditions:
1. e ∈ C1;
2. (i) Mn(B) = (Cn − Cn) + i(Cn − Cn), for all n ∈ N,
(ii) Cn ∩ (−Cn) = {0}, for all n ∈ N,
(iii) (Cn − Cn) ∩ i(Cn − Cn) = {0}, for all n ∈ N;
3. (i) for all c1, c2 ∈ Cn and c ∈ Cn, we have that (c1−c2)c(c1−c2) ∈ Cn,
(ii) for all n, m and B ∈Mn×m(C) we have that B
∗CnB ⊆ Cm;
4. there is r > 0 such that for every positive integer n and c ∈ Cn − Cn
we have r‖c‖en + c ∈ Cn,
5. there exists a constant K > 0 such that for all n ∈ N and a, b ∈ Cn−Cn
we have ‖a‖n ≤ K · ‖a+ ib‖n.
Theorem 4. If an operator algebra B has a ∗-admissible sequence of cones
then there is a completely bounded isomorphism τ from B onto a C∗-algebra
A. If, in addition, one of the following conditions holds
(1) there exists r > 0 such that for every n ≥ 1 and c, d ∈ Cn we have
‖c+ d‖ ≥ r‖c‖.
(2) ‖(x− iy)(x+ iy)‖ ≥ α‖x− iy‖‖x+ iy‖ for all x, y ∈ Cn − Cn
then the inverse τ−1 : A → B is also completely bounded.
Conversely, if such isomorphism τ exists then B possesses a ∗-admissible
sequence of cones and conditions (1) and (2) are satisfied.
The proof will be divided into 4 lemmas.
Let {Cn}n≥1 be a ∗-admissible sequence of cones of B. Let B2n =M2n(B),
φn : B2n → B2n+1 be unital homomorphisms given by φn(x) =
x ∈ B2n . Denote by B∞ = lim−→
B2n the inductive limit of the system (B2n , φn).
As all inclusions φn are unital B∞ has a unit, denoted by e∞. Since B∞ can
be considered as a subalgebra of a C∗-algebra of the corresponding induc-
tive limit of M2n(B(H)) we can define the closure of B∞ in this C
∗-algebra
denoted by B∞.
Now we will define an involution on B∞. Let ξn : M2n(B) → B∞ be the
canonical morphisms. By (3ii), φn(C2n) ⊆ C2n+1 . Hence C =
ξn(C2n) is a
well defined cone in B∞. Denote by C its completion. By (2i) and (2iii), for
every x ∈ B2n , we have x = x1+ ix2 with unique x1, x2 ∈ C2n −C2n . By (3ii)
we have
∈ C2n+1 − C2n+1, i = 1, 2. Thus for every x ∈ B∞ we
have unique decomposition x = x1+ ix2, x1 ∈ C−C, x2 ∈ C−C. Hence the
mapping x 7→ x♯ = x1− ix2 is a well defined involution on B∞. In particular,
we have an involution on B which depends only on the cone C1.
Lemma 5. Involution on B∞ is defined by the involution on B, i.e. for all
A = (aij)i,j ∈M2n(B)
A♯ = (a
ji)i,j.
Proof. Assignment A◦ = (a
ji)i,j, clearly, defines an involution on M2n(B).
We need to prove that A♯ = A◦.
Let A = (aij)i,j ∈ M2n(B) be self-adjoint A
◦ = A. Then A =
aii ⊗
Eii +
(aij ⊗ Eij + a
ij ⊗ Eji) and a
ii = aii, for all i. By (3ii) we have
aii ⊗ Eii ∈ C2n − C2n . Since aij = a
ij + ia
ij for some a
ij , a
ij ∈ C2n − C2n
we have
aij ⊗Eij + a
ij ⊗ Eji = (a
ij + ia
ij)⊗ Eij + (a
ij − ia
ij)⊗ Eji
= (a′ij ⊗ Eij + a
ij ⊗ Eji) + (ia
ij ⊗ Eij − ia
ij ⊗ Eji)
= (Eii + Eji)(a
ij ⊗ Eii + a
ij ⊗ Ejj)(Eii + Eij)
− (a′ij ⊗ Eii + a
ij ⊗ Ejj)
+ (Eii − iEji)(a
ij ⊗ Eii + a
ij ⊗ Ejj)(Eii + iEij)
− (a′′ij ⊗ Eii + a
ij ⊗ Ejj) ∈ C2n − C2n .
Thus A ∈ C2n − C2n and A
♯ = A. Since for every x ∈ M2n(B) there exist
unique x1 = x
1 and x2 = x
2 in M2n(B), such that x = x1 + ix2, and unique
x′1 = x
1 and x
2 = x
2 , such that x = x
1 + ix
2, we have that x1 = x
1 = x
x2 = x
2 = x
2 and involutions ♯ and ◦ coincide.
Lemma 6. Involution x → x♯ is continuous on B∞ and extends to the in-
volution on B∞. With respect to this involution C ⊆ (B∞)sa and x
♯Cx ⊆ C
for every x ∈ B∞.
Proof. Consider a convergent net {xi} ⊆ B∞ with the limit x ∈ B∞. Decom-
pose xi = x
i with x
i ∈ C−C. By (5), the nets {x
i} and {x
i } are also
convergent. Thus x = a+ ib, where a = lim x′i ∈ C − C, b = lim x
i ∈ C − C
and lim x
i = a− ib. Therefore the involution defined on B∞ can be extended
by continuity to B∞ by setting x
♯ = a− ib.
Under this involution C ⊆ (B∞)sa = {x ∈ B∞ : x = x
Let us show that x♯cx ∈ C for every x ∈ B∞ and c ∈ C. Take firstly
c ∈ C2n and x ∈ B2n . Then x = x1 + ix2 for some x1, x2 ∈ C2n − C2n and
(x1 + ix2)
♯c(x1 + ix2) = (x1 − ix2)c(x1 + ix2)
)( −x1 −ix2
ix2 x1
−x1 −ix2
ix2 x1
By (3i), Lemma 5 and (3ii) x♯cx ∈ C2n .
Let now c ∈ C and x ∈ B∞. Suppose that ci → c and xi → x, where
ci ∈ C, xi ∈ B∞. We can assume that ci, xi ∈ B2ni . Then x
icixi ∈ C2ni for
all i and since it is convergent we have x♯cx ∈ C.
Lemma 7. The unit of B∞ is an Archimedean order unit and (B∞)sa =
C − C.
Proof. Firstly let us show that e∞ is an order unit. Clearly, (B∞)sa = C − C.
For every a ∈ C − C, there is a net ai ∈ C2ni − C2ni convergent to a. Since
‖ai‖ <∞ there exists r1 > 0 such that r1eni − ai ∈ C2ni , i.e. r1e∞− ai ∈
C. Passing to the limit we get r1e∞ − a ∈ C. Replacing a by −a we can
find r2 > 0 such that r2e∞ + a ∈ C. If r = max(r1, r2) then re∞ ± a ∈ C.
This proves that e∞ is an order unit and that for all a ∈ C − C we have
a = re∞ − c for some c ∈ C. Thus C − C ∈ C − C. The converse inclusion,
clearly, holds. Thus C − C = C − C.
If x ∈ (B∞)sa such that for every r > 0 we have r + x ∈ C then x ∈ C
since C is closed. Hence e∞ is an Archimedean order unit.
Lemma 8. B∞ ∩ C = C.
Proof. Denote by D = lim
M2n(B(H)) the C
∗-algebra inductive limit corre-
sponding to the inductive system φn and denote φn,m = φm−1 ◦ . . . ◦ φn :
M2n(B(H)) → M2m(B(H)). For n < m we identify M2m−n(M2n(B(H)))
with M2m(B(H)) by omitting superfluous parentheses in a block matrix
B = [Bij ]ij with Bij ∈M2n(B(H)).
Denote by Pn,m the operator diag(I, 0, . . . , 0) ∈M2m−n(M2n(B(H))) and
set Vn,m =
∑2m−n
k=1 Ek,k−1. Here I is the identity matrix in M2n(B(H)) and
Ek,k−1 is 2
n×2n block matrix with identity operator at (k, k−1)-entry and all
other entries being zero. Define an operator ψn,m([Bij ]) = diag(B11, . . . , B11).
It is easy to see that
ψn,m([Bij ]) =
2m−n−1∑
(V kn,mPn,m)B(V
n,mPn,m)
Hence by (3ii)
ψn,m(C2m) ⊆ φ(C2n) ⊆ C2m . (4)
Clearly, ψn,m is a linear contraction and
ψn,m+k ◦ φm,m+k = φm,m+k ◦ ψn,m
Hence there is a well defined contraction ψn = lim
ψn,m : D → D such that
ψn|M2n(B(H)) = idM2n (B(H)),
whereM2n(B(H)) is considered as a subalgebra in D. Clearly, ψn(B∞) ⊆ B∞
and ψn|B2n = id. Consider C and C2n as subalgebras in B∞, by (4) we have
ψn : C → C2n.
To prove that B∞ ∩C = C take c ∈ B∞ ∩C. Then there is a net cj in C
such that ‖cj − c‖ → 0. Since c ∈ B∞, c ∈ B2n for some n, and consequently
ψn(c) = c. Thus
‖ψn(cj)− c‖ = ‖ψn(cj − c)‖ ≤ ‖cj − c‖.
Hence ψn(cj) → c. But ψn(cj) ∈ C2n and the latter is closed. Thus c ∈ C.
The converse inclusion is obvious.
Remark 9. Note that for every x ∈ D
ψn(x) = x. (5)
Indeed, for every ε > 0 there is x ∈ M2n(B(H)) such that ‖x − xn‖ < ε.
Since ψn is a contraction and ψn(xn) = xn we have
‖ψn(x)− x‖ ≤ ‖ψn(x)− xn‖+ ‖xn − x‖
= ‖ψn(x− xn)‖+ ‖xn − x‖ ≤ 2ε.
Since xn ∈ M2n(B(H)) also belong to M2m(B(H)) for all m ≥ n, we have
that ‖ψm(x)− x‖ ≤ 2ε. Thus lim
ψn(x) = x.
Proof of Theorem 4. By Lemma 6 and 7 the cone C and the unit
e∞ satisfies all assumptions of Theorem 1. Thus there is a homomorphism
τ : B∞ → B(H̃) such that τ(a
♯) = τ(a)∗ for all a ∈ B∞. Since the image
of τ is a ∗-subalgebra of B(H̃) we have that τ is bounded by [3, (23.11),
p. 81]. The arguments at the end of the proof of Theorem 2 show that the
restriction of τ to B2n is unitary equivalent to the 2
n-amplification of τ |B.
Thus τ |B is completely bounded.
Let us prove that ker(τ) = {0}. By Theorem 2.3 it is sufficient to show
that C ∩ (−C) = 0. If c, d ∈ C such that c + d = 0 then c = d = 0. Indeed,
for every n ≥ 1, ψn(c) + ψn(d) = 0. By Lemma 8, we have
ψn(C) ⊆ C ∩ B2n = C2n .
Therefore ψn(c), ψn(d) ∈ C2n . Hence ψn(c) = −ψn(d) ∈ C2n ∩ (−C2n) and,
consequently, ψn(c) = ψn(d) = 0. Since ‖ψn(c)−c‖ → 0 and ‖ψn(d)−d‖ → 0
by Remark 9, we have that c = d = 0. If x ∈ C ∩ (−C) then x+ (−x) = 0,
x,−x ∈ C and x = 0. Thus τ is injective.
We will show that the image of τ is closed if one of the conditions (1) or
(2) of the statement holds.
Assume firstly that operator algebra B satisfies the first condition. Since
τ(B∞) = τ(C)−τ(C)+ i(τ(C)−τ(C)) and τ(C) is exactly the set of positive
operators in the image of τ , it is suffices to prove that τ(C) is closed. By
Theorem 1.3, for self-adjoint (under involution ♯) x ∈ B∞ we have
‖τ(x)‖
B( eH)
= inf{r > 0 : re∞ ± x ∈ C}.
If τ(cα) ∈ τ(C) is a Cauchy net in B(H̃) then for every ε > 0 there is γ
such that ε ± (cα − cβ) ∈ C when α ≥ γ and β ≥ γ. Since C ∩ B∞ = C,
ε ± (cα − cβ) ∈ C. Denote cαβ = ε + (cα − cβ) and dαβ = ε − (cα − cβ).
The set of pairs (α, β) is directed if (α, β) ≥ (α1, β1) iff α ≥ α1 and β ≥ β1.
Since cαβ + dαβ = 2ε this net converges to zero in the norm of B∞. Thus by
assumption 4 in the definition of ∗-admissible sequence of cones, ‖cαβ‖B∞ →
0. This implies that cα is a Cauchy net in B∞. Let c = lim cα. Clearly,
c ∈ C. Since τ is continuous ‖τ(cα) − τ(c)‖B∞ → 0. Hence the closure
τ(C) is contained in τ(C). By continuity of τ we have τ(C) ⊆ τ(C). Hence
τ(C) = τ(C), τ(C) is closed.
Let now B satisfy condition (2) of the Theorem. Then for every x ∈ B∞
we have ‖x♯x‖ ≥ α‖x‖‖x♯‖. By [3, theorem 34.3] B∞ admits an equivalent
C∗-norm |·|. Since τ is a faithful ∗-representation of the C∗-algebra (B∞, |·|)
it is isometric. Therefore τ(B∞) is closed.
Let us show that (τ |B)
−1 : τ(B) → B is completely bounded. The image
A = τ(B∞) is a C
∗-algebra inB(H̃) isomorphic to B∞. By Johnson’s theorem
(see [6]), two Banach algebra norms on a semi-simple algebra are equivalent,
hence, τ−1 : A → B∞ is bounded homomorphism, say ‖τ
−1‖ = R. Let us
show that ‖(τ |B)
−1‖cb = R. Since
τ |B2n = Un(τ |B ⊗ idM2n )U
for some unitary Un : K ⊗ C
2n → H̃ we have for any B = [bij ] ∈M2n(B)
bij ⊗ Eij‖ ≤ R‖τ(
bij ⊗ Eij)‖
= R‖Un(
τ(bij)⊗ Eij)U
τ(bij)⊗ Eij‖.
This is equivalent to
τ−1(bij)⊗ eij‖ ≤ R‖
bij ⊗Eij‖,
hence ‖(τ−1)2
(B)‖ ≤ R‖B‖. This proves that ‖(τ |B)
−1‖cb = R.
The converse statement evidently holds with ∗-admissible sequence of
cones given by (τ (n))−1(Mn(A)
Conditions (1) and (2) were used to prove that the image of isomorphism τ
is closed. The natural question one can ask is wether there exists an operator
algebra B and isomorphism ρ : B → B(H) with non-closed self-adjoint image.
The following example gives the affirmative answer.
Example 10. Consider the algebra B = C1([0, 1]) as an operator algebra in
C∗-algebra
M2(C([0, 1])) via inclusion
f(·) 7→ ⊕q∈Q
f(q) f ′(q)
0 f(q)
The induced norm
‖f‖ = sup
(2|f(q)|2 + |f ′(q)|2 + |f ′(q)|
4|f(q)|2 + |f ′(q)|2)
satisfies the inequality ‖f‖ ≥ 1√
max{‖f‖∞, ‖f
′‖∞} ≥
‖f‖1 where ‖f‖1 =
‖f‖∞+‖f
′‖∞ is the standard Banach norm on C
1([0, 1]). Thus B is a closed
operator algebra with isometric involution f ♯(x) = f(x), (x ∈ [0, 1]). The
identity map C1([0, 1]) → C([0, 1]), f 7→ f is a ∗-isomorphism of B into
C∗-algebra with non-closed self-adjoint image.
4 Operator Algebra associated with Kadison’s
similarity problem.
In 1955 R. Kadison raised the following problem. Is any bounded homomor-
phism π of a C∗-algebra A into B(H) similar to a ∗-representation? The
similarity above means that there exists invertible operator S ∈ B(H) such
that x→ S−1π(x)S is a ∗-representation of A.
The following criterion due to Haagerup (see [4]) is widely used in refor-
mulations of Kadison’s problem: non-degenerate homomorphism π is similar
to a ∗-representation iff π is completely bounded. Moreover the similarity S
can be chosen in such a way that ‖S−1‖‖S‖ = ‖π‖cb.
The affirmative answer to the Kadison’s problem is obtained in many
important cases. In particular, for nuclear A, π is automatically completely
bounded with ‖π‖cb ≤ ‖π‖
2 (see [1]).
About recent state of the problem we refer the reader to [9, 5].
We can associate an operator algebra π(B) to every bounded injective
homomorphism π of a C∗-algebra A. The fact that π(B) is closed can be seen
by restricting π to a nuclear C∗-algebra C∗(x∗x). This restriction is similar to
∗-homomorphism for every x ∈ A which gives the estimate ‖x‖ ≤ ‖π‖3‖π(x)‖
(for details see [10, p. 4]). Denote Cn = π
(n)(Mn(A)
Let J be an involution in B(H), i.e. self-adjoint operator such that J2 =
I. Clearly, J is also a unitary operator. A representation π : A → B(H) of a
∗-algebra A is called J-symmetric if π(a∗) = Jπ(a)∗J . Such representations
are natural analogs of ∗-representations for Krein space with indefinite metric
[x, y] = 〈Jx, y〉.
We will need the following observation due to V. Shulman [13] (see also
[7, lemma 9.3, p.131]). If π is an arbitrary representation of A in B(H) then
the representation ρ : A → B(H⊕H), a 7→ π(a)⊕π(a∗)∗ is J-symmetric with
J(x⊕ y) = y⊕x and representation π is a restriction ρ|K⊕{0}. Moreover, if ρ
is similar to ∗-representation then so is π. Clearly the converse is also true,
thus π and ρ are simultaneously similar to ∗-representations or not. In sequel
for an operator algebra D ∈ B(H) we denote by lim
M2n(D) the closure of the
algebraic direct limit of ofM2n(D) in the C
∗-algebra direct limit of inductive
system M2n(B(H)) with standard inclusions x→
Theorem 11. Let π : A → B(H) be a bounded unital J-symmmetric in-
jective homomorphism of a C∗-algebra A and let B = π(A). Then π−1 is
a completely bounded homomorphism. Its extension π̃−1 to the homomor-
phism between the inductive limits B∞ = lim−→
M2n(B) and A∞ = lim−→
M2n(A)
is injective.
Proof. Let us show that {Cn}n≥1 is a ∗-admissible sequence of cones. It
is routine to verify that conditions (1)-(3) in the definition of ∗-admissible
cones are satisfied for {Cn}. To see that condition (4) also holds take B ∈
Cn − Cn and denote r = ‖B‖. Let D ∈ Mn(A)sa be such that B = π
(n)(D).
Since π(n) : Mn(A) → Mn(B) is algebraic isomorphism it preserves spectra
σMn(A)(x) = σMn(B)(π
(n)(x)). Since the spectral radius spr(B) ≤ r we have
spr(D) ≤ r. Hence ren +D ∈ Mn(A)
+ because D is self-adjoint. Applying
π(n) we get ren +B ∈ Cn which proves condition (4).
Since π is J-symmetric
‖π(n)(a)‖ = ‖(J ⊗En)π
(n)(a)∗(J ⊗En)‖ = ‖π
(n)(a∗)‖
for every a ∈Mn(A), and
‖π(n)(h1)‖ ≤ 1/2(‖π
(n)(h1) + iπ
(n)(h2)‖+ ‖π
(n)(h1)− iπ
(n)(h2)‖)
= ‖π(n)(h1) + iπ
(n)(h2)‖
for all h1, h2 ∈ Cn − Cn. Thus condition (5) is satisfied and {Cn} is ∗-
admissible. By Theorem 4, there is an injective bounded homomorphism
τ : B∞ → B(H̃) such that its restriction to B is completely bounded, τ(b
τ(b)∗ and τn(Cn) = τn(Mn(B))
Denote ρ = τ ◦ π : A → B(H̃). Since ρ is a positive homomorphism, it is
a ∗-representation. Moreover, ker ρ = {0} because both π and τ are injective.
Therefore ρ−1 is ∗-isomorphism. Since τ : B → B(H̃) extends to an injective
homomorphism of inductive limit B∞ and ρ
−1 is completely isometric, we
have that π−1 = ρ−1 ◦ τ extends to injective homomorphism of B∞. It is also
clear that π−1 is completely bounded as a superposition of two completely
bounded maps.
Remark 12. The first statement of Theorem 11 can be deduced also from [10,
Theorem 2.6].
Remark 13. Note that condition (1) and (2) in Theorem 4 for cones Cn
from the proof of Theorem 11 is obviously equivalent to π being completely
bounded.
Acknowledgments.
The authors wish to express their thanks to Victor Shulman for helpful
comments and providing the reference [13].
The work was written when the second author was visiting Chalmers
University of Technology in Göteborg, Sweden. The second author was sup-
ported by the Swedish Institute.
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	Introduction
	Operator realizations of matrix-ordered *-algebras. 
	Operator Algebras completely boundedly isomorphic to C*-algebras.
	Operator Algebra associated with Kadison's similarity problem.
