On Kadison’s Similarity Problem for Homomorphisms of the Algebra of Complex Polynomials ()
1. Introduction
In 1950, Dixmier established the following similarity problem: Whether or not bounded representations of groups are necessarily similar to unitary representations. The same year, Dixmier proved that every uniformly bounded representation of an amenable group G on a Hilbert space H is similar to a unitary representation of G on H [1]. However, in 1955, L. Ehrenpreis and F. Mautner [2] gave examples showing that this result fails when the amenability assumption is removed. Therefore, there exist bounded homomorphisms of groups into the group of invertible operators on a Hilbert space, which are not similar to unitary representations of the group. In 1955, motivated by Dixmier’s similarity problem, the analogous problem for representations of C*-algebras was introduced by Kadison. Kadison’s similarity problem asks whether every bounded unital representation of a unital C*-algebra
on a Hilbert space H is similar to a *-representation of
on H [3]. In 1960, Kunze-Stein proved that Dixmier’s question has a negative answer, that is, there exist bounded representations of groups
that are not similar to unitary representations [4]. In the early 1980’s, motivated by the work of Wittstock and Haagerup, the theory of completely positive maps was extended to the family of completely bounded maps by many researchers [5]. The theory of completely bounded maps has many deep connections to similarity questions. In 1981, Christensen solves the original problem for irreducible representations (so, each vector is cyclic for the image) [6]. In 1983, Haagerup gave a positive answer to Kadison’s similarity problem for cyclic representations and proved that a bounded representation π of a C*-algebra
on H is similar to a *-representation if and only if π is completely bounded. The result of Haagerup shows that Kadison’s similarity problem is equivalent to determining whether or not bounded homomorphisms are necessarily completely bounded. There are several important cases where bounded homomorphisms of a C*-algebra are completely bounded. Haagerup proves a number of other results. For example, he settles the problem completely for C*-algebras that admit no trace. He observes that the problem reduces to studying representations of von Neumann algebras. Thus the problem is settled positively for von Neumann algebras with no central summand of finite type [7]. In 1984, Vern Paulsen generalized Haagerup’s result to the non-self-adjoint case. He proved that every unital homomorphism of an operator algebra is a completely bounded homomorphism if and only if it is similar to a completely contractive homomorphism [8]. However, contractive homomorphisms need not be completely contractive. Parrott introduced in [9] an example of a contractive homomorphism which is not a completely contractive homomorphism. In 1986, Christensen settles the problem for factors of type
with property
[10]. We say that the C*-algebra
has the similarity property if the similarity problem has a positive answer for
. In 1996, Kirchberg showed that Kadison’s similarity problem is equivalent to another important open question, the derivation problem, itself a crucial problem in the cohomology theory of operator algebras [11]. In 1999, motivated by the similarity problem, Pisier introduced the notion of the length
of an operator algebra
in [12] and examined its properties in [13] [14]. This integer arises from the ability to write matrices over
as products of bounded length, where the constituent factors alternate between scalar matrices and diagonal matrices over
. If such decompositions do not exist then
. An easy consequence of finite length is that all bounded homomorphism of
into any B(H) are completely bounded, which solves the similarity problem for such algebras, and is indeed equivalent to it. In 2010, Christensen, Sinclair, Smith and White examine connections between the theory of perturbations and Kadison’s similarity problem [15]. At the present time, the similarity problem is still open. The Factorization of matrices of complex polynomials of several variables, in terms of products of Matrices of polynomials of two variables with positive matrices as Fourier coefficients, gives us some light on von Neumann’s inequality for matrices of complex polynomials of several variables.
Kadison’s similarity problem has many engineering applications.
In this paper, we prove that homomorphisms of the algebra
are completely bounded. We show that Parrott’s contractive homomorphism, which is not completely contractive, is completely bounded (similar to a completely contractive homomorphism). We also prove that every homomorphism of the algebra
generates completely positive maps over the algebras
and
.
2. Main Result
In this section, we prove our main result and we show that Parrott’s contractive homomorphism is similar to a completely contractive homomorphism over the algebra of complex polynomials over
. We construct completely positive maps over the algebras
and
. Given
a complex polynomial over
. Then
Let
denote the algebra of complex polynomials over
. Given
complex polynomials over
. Then
Denote by
Then the
-matrix
can be written as
The
-matrix
can be regarded as a polynomial over
with matrix coefficients. In this case, the set
is the set of Fourier coefficients of
. Let us set
where the supremum is taken over the family of all n-tuples of commuting contractions on all Hilbert spaces. It is easy to see that
is finite, since it is bounded by the sum of the norm operators of the Fourier coefficients off, and that this quantity defines a norm on the algebra
of matrices of polynomials over
. For each polynomial P in
, there is always an n-tuple of contractions where this supremum is achieved. Therefore,
and
are normed algebras. Let
be a matrix of complex polynomials over
. Define a function
by setting
We can check that
defines a norm on
. Therefore,
is also a normed algebra.
The Hahn-Banach extension property states that a vector subspace Y of X has the extension property if any continuous linear functional on Y can be extended to a continuous linear functional on X. The Hahn-Banach theorem is a useful tool in Functional Analysis. It allows the extension of bounded linear functional defined on a subspace of some vector space to the whole space. The proof of the Hahn-Banach type extension theorem for completely bounded maps introduced by Wittstock can be found in [5].
Theorem 2.1. (Wittstock’s Extension Theorem). Let
be a C*-algebra,
a subspace of
, and let
be completely bounded. Then there exists a completely bounded map
which extends
, with
.
Kadison’s similarity problem is that every bounded homomorphism of a C*-algebra into B(H) is similar to a *-homomorphism. Haagerup proved that every unital bounded homomorphism of a C*-algebra is similar to a *-homomorphism if and only if it is a completely bounded homomorphism [7].
Theorem 2.2. [7]. Let
be a C*-algebra with unit and let
be a bounded, unital homomorphism. Then
is similar to a *-homomorphism if and only if
is completely bounded. Moreover, if
is completely bounded, then there exists a similarity S with
is a *-homomorphism and
Haagerup’s result shows that Kadison’s similarity problem is equivalent to determining whether or not bounded homomorphisms are necessarily completely bounded. Paulsen generalized Haaerup’s result to the non-self-adjoint case and he proved several results on completely bounded maps over unital C*-algebras [8].
Theorem 2.3. [5]. Let
be an operator algebra and let
be unital completely bounded homomorphism. Then there exists an invertible operator S with
such that
is a completely contractive homomorphism. Moreover,
Theorem 2.4. Let
be a C*-algebra with unit and let
be completely bounded. Then there exist completely positive maps
with
such that the map
given by
is completely positive. Moreover, if
, then we may take
The structure of the Fourier coefficients of a complex polynomial
over
is linked to the properties of
. For example, if
is a complex polynomial with a positive real number as Fourier coefficients, then
In this case,
for any n-tuple
of commuting contractions on a Hilbert space. Complex polynomials over
with positive real numbers as Fourier coefficients allow the construction of complex polynomials over
.
Remark 2.5. Let
be a complex polynomial over
. Then
there exists a constant
,
, such that
Let
, be complex polynomials over
. Then there exists a complex polynomial
over
such that
It is possible to factorize matrices of complex polynomials. Let
be a matrix of complex polynomials of three variables of
. Then
there exists a constant
,
, such that
Let us notice that
is a positive matrix of complex polynomials and
is not unique. Denote by
and
It is straightforward to see that
This implies that there exists a matrix
of complex polynomials over
such that
In general, matrices of polynomials over
with positive matrices as Fourier coefficients allow the construction of matrices of complex polynomials over
.
Remark 2.6. Let
be a matrix of complex polynomials over
. Then
there exists a constant
,
, such that
Let
, be matrices of complex polynomials over
. Then there exists a matrix of complex polynomials
over
such that
What we need to notice is that the sets
are not bounded. However, the factorization of matrices of complex polynomials, in terms of the product of matrices of complex polynomials of two variables, did allows us to prove that the set
is bounded by
.
Theorem 2.7. Let
be the algebra of complex polynomials over
and let
be a set of commuting contractions on the Hilbert space H. Then
Proof. Let
be a matrix of complex polynomials over
. Then
there exists a constant
,
, such that
Recall that
Denote by
the matrix of complex polynomials with positive matrices as Fourier coefficients. The matrix of complex polynomials
satisfies the von Neumann’s inequality. Denote by
and
. As we can see
, are matrices of complex polynomials of two variables with positive matrices as Fourier coefficients and
, are matrices of complex polynomials of one variable with positive matrices as Fourier coefficients. These polynomials have exactly the same number of Fourier coefficients than
and they all satisfy the von Neumann inequality. It is straightforward to see that
and
Denote by
That is,
. Namelly,
It is not difficult to see that
This implies that there exists a matrix
of complex polynomials over
such that
In other words,
It is straightforward to see that
In general, the matrix
is not unique. The structure of the matrix
depends on the structure of
. Let
be an n-tuple of commuting contractions on a Hilbert space H. The von Neumann inequality for complex polynomials of two (or one) variable(s) allows us to say that
This implies that
Suppose that
It follows that
This implies that,
Thus,
Also, we have
It is easy to say that
Therefore,
We can claim that
since, in general,
It is clear that
This implies that
since
. Due to the fact that
and
we can say that for every
, there exist two positive real numbers
and
such that
and
Thus,
and
Therefore,
It is not difficult to see that
Now, we can claim that
The Archimedean property of the total order on
allows us to say that there exists a positive real number
such that
where
is the smallest positive integer such that
This allows us to say that
It follows that
since
. In other words,
with
. Let us notice that
(2.1)
This implies that
This means that
(2.2)
Also, we can notice that
(2.3)
It is straightforward to observe that if we add the equations (2.1) and (2.3), one has
(2.4)
We can claim now that
It follows that
(2.5)
We can say that the set
is bounded. The set
is also bounded. Indeed, assume that there exists
such that
This implies that
We have a contradiction because
Therefore, there exists a positive constant
such that
Suppose that
It is clear that
. Let us estimate
. Assume that
. One has
. Now, assume that
. We have
. Therefore, for every
, we have
(2.6)
The equation (2.6) allows us to say that for every
, we have
(2.7)
In particular, if we choose the matrix
, one has
(2.8)
In other words,
Finally,
Let us observe that if
then for every
, one has
In particular, for
, one has
In fact,
. Theorem 2.7, Theorem 2.1 and Theorem 2.2 allow us to prove our main result. Every homomorphism of the algebra
is completely bounded.
Theorem 2.8. Let
be the algebra of complex polynomials over
, let
be a set of commuting contractions on the Hilbert space H and let
be the map given by
Then there exists a homomorphism
similar to a *-homomorphism such that
Proof. Let
be the algebra of complex polynomials over
and let
be a set of commuting contractions on the Hilbert space H. We just need to show that the map
given by
is a completely bounded homomorphism. It is well known that this map is a homomorphism. Let us show that the map
given by
is bounded for all
with
Theorem 2.7 allows us to claim that
It is well known that
In other words,
It follows that
Therefore,
Finally, the homomorphism
is completely bounded. Theorem 2.1 enables us to say that there exists a completely bounded homomorphism
such that
Theorem 2.2 allows us to claim that the homomorphism
is similar to a *-homomorphism. In other words, there exists an invertible operator
such that
is a *-homomorphism. In particular,
In general, the properties of every homomorphism of the algebra
depend on the structure of the associated tuple
which generates that homomorphism. For instance, if the tuple
is a doubly commuting set of commuting contractions (or circulant contractions, complex triangular Toeplitz contractions) the corresponding homomorphism
given by
is completely contractive [16] , since von Neumann’s inequality hold for this type of tuples.
Theorem 2.8 and Theorem 2.3 allow us to show that the contractive homomorphism introduced by Parrott in 1970, which is not completely contractive, is similar to a completely contractive homomorphism (completely bounded).
Proposition 2.9. Let U and V be contractions in B(H) such that U is unitary and U and V don’t commute. We define commuting contraction on
by setting
Let
be the algebra of complex polynomials over
, let
be a set of commuting contractions on the Hilbert space H. Then the homomorphism
given by
is a completely bounded homomorphism. Moreover,
.
Proof. Theorem 2.8 and Theorem 2.3 allow us to claim that the homomorphism
is a completely bounded homomorphism, similar to a completely contractive homomorphism and
We can now introduce the existence of completely positive maps over the C*-algebras
and
. Every homomorphism of the algebra
generates completely positive maps over the algebras
and
.
Theorem 2.10. Let
be a set of commuting contractions on the Hilbert space H and let
be the homomorphim given by
. Then there exist completely positive maps
with
such that the map
given by
is completely positive.
Proof. Theorem 2.8 and Theorem 2.2 allow us to claim that homomorphism
is completely bounded. Theorem 2.1 enables us to say that there exists a completely bounded homomorphism
such that
Theorem 2.4 yields the desired result.