On Maps Preserving the Spectrum on Positive Cones of Operator Algebras ()
1. Introduction
In the research field of operator algebra, preserving problem is one of the hot research directions. Over the years, a large number of mathematicians and researchers have devoted themselves to the study of it ( [1] [2] [3]). There are also many works on preserves of Kubo-Ando means by M. Gaál, G. Nagy, Lei Li, Molnár L., Semrl P. and Liguang Wang ( [4] - [12]). These preserves are characterized by Jordan *-isomorphisms.
We now recall the concept and properties of Jordan
-isomorphisms that will be used in this paper. We refer to [3] for more properties of Jordan
-isomorphisms. Suppose that
and
are
-algebras, a bijective linear map
is a Jordan
-isomorphism if it satisfies
for all
. The Jordan
-isomorphism
preserves the Jordan triple product, i.e., for any
, we have that
For any
,
is invertible if and only if
is invertible and, moreover,
In particular,
preserves the spectrum of the elements, i.e.,
for any
. We also have
for any
and continuous real function
on its spectrum. Moreover,
preserves commutativity in both directions, i.e., for any
, we have
Finally, for any
we have
and
is an isometry,
Let
be
-algebras with the set of all self-adjoint
respectively. Suppose
be a surjective map. Molnár in [13] considered the map that satisfies
(1.1)
and
(1.2)
He showed that these maps are characterized by Jordan *-isomorphism.
In this paper, we would like to consider when there are two maps in (1.1) and (1.2). The results obtained in this generalize Molnár’s works in [13]. Suppose
and
be surjective maps. Now we consider the following structures
and
In the process of proving the theorem, we describe some lemmas, and then give the results and proofs.
2. Main Results
Now we first give fives lemmas that will use in the proving theorems.
Lemma 2.1. ( [14]) Let
be
-algebras. If is a surjective linear isometry, then it of the form
where
is a unitary element and is a Jordan
-isomorphism.
Lemma 2.2. ( [7]) Let
be
-algebras and let
be a bijective linear map which preserves the order in both direction, i.e., satisfies
Then
is of the form
where
is an invertible element and
is a Jordan
-isomorphism.
Lemma 2.3. ( [13]) Let
be a von Neumann algebra. Assume
is a symmetry such that for every symmetry
, the spectrum
contains only real numbers. Then
is a central symmetry in
.
Lemma 2.4. ( [13]) Let
be a von Neumann algebra. Pick
. If
holds for all
, then we have
.
Lemma 2.5. ( [9]) Let
be a
-algebra and pick
, then
The following are the main results obtained in this paper.
Theorem 2.1. Suppose
and
are
-algebras and
are surjective maps. Then
and
if and only if there is a Jordan isomorphism
such that
.
Proof. (
) Assume that
Then we have
i.e.,
and
. Hence
. For any
, we have
Thus
for all
and
is a surjective isometry. It follows from Mazur-Ulam theorem ( [15]) that
is a linear surjective isometry. Then by Lemma 2.1 there exist a Jordan isomorphism
and a center symmetry
such that
By the spectrum-preserving property of
, put
,
, we infer
and therefore
. Since
, we have
. Hence
Since
we have
for
.
(
) If
, then we have
This completes the proof. □
In the proof of next theorem we also need the concept of the Thompson metric (see [16]) which we denote by
. Define
where
for
. It also holds:
By Theorem 9 in [16], for every such isometry
, there is a Jordan
-isomorphism
, a center projection
and a positive invertible element
such that
Theorem 2.2. Let
and
be von Neumann algebras. Suppose
be surjective maps. Then
and
if and only if there exists a Jordan isomorphism
such that
Proof. (
) Suppose
and
satisfy
and
. Let
and we have
This implies that
. By Lemma 2.3,
is a central symmetry. Then it follows that
and
preserve the spectrum. In particular, we obtain that
and
maps
onto
. For any
, we have
and this implies that
, i.e.,
. If
, we get
By the definition of Thompson metric, this implies that
is a surjective Thompson isometry from
onto
.
Since for all
and all real numbers
, we have
It follows from Lemma 2.4 that
is also homogeneous. Hence
is a Thompson isometry. By Theorem 9 in [16] that there exist a Jordan isomorphism
such that
holds for all
. We need to show that
for all
.
If
, then we have
For any
and arbitrary
, we have
and then Lemma 2.4 implies that
, i.e.,
,
. For any
and
, we have
Hence
and
for all
.
(
) For
, we have
This completes the proof. □
Let
be a standard
-algebra acting on a Hilbert space
. Suppose
, such that
holds for all
. Pick
and
, we have
. By the Corollary 3.4.5 in [17],
holds for every rank-one projective
. Then for every unit vector
in
, we have
and therefore
. Now we can prove the following result.
Theorem 2.3. Let
be standard
-algebras. Suppose
is acting on the Hilbert space
and
is acting on the Hilbert space
. Let
be surjective maps. Then
and
if and only if there exist a constant
and either a unitary or an antiunitary operator
such that
Proof. (
) Suppose that
are surjective maps with
For vectors
in
, we denote the rank one operator
defined by
,
. We show that
is injective. If
, such that
. Then for every
that
For
, where
is an arbitrary vector and
, then we have
Thus
and therefore
. For the self-adjoint element
, we have
thus
and
is a symmetry.
Now we show that
. Since
so
is real for every
. Suppose
. Then there is a basis
in the Hilbert space
such that
where neither of these sums is zero. Pick
and
. Define
Then
and
, that’s a contradiction. It follows that
. If
, we have
for every
, and
maps
onto
. Thus
is a surjective Thompson isomorphism from
onto
. It follows that there is a Jordan
-isomorphism
such that
By the result of Herstein [10],
is either a
-isomorphism or a
-antiisomorphism. And from [3], we have either a unitary operator
such that
or an antiunitary operator
such that
The map
satisfies
and it equals the identity on
. If
, we have
So
, i.e.,
. For any
, we get
Therefore,
.
(
) For
, we have
This completes the proof. □
The following characterization of the order of operators is needed in the proof of Theorem 2.4.
Theorem 2.4. Let
and
be two
-algebras and
be a surjective map. Then
if and only if there is a Jordan
-isomorphism
such that
for every
.
Proof. (
) Suppose
is a surjective map satisfies
First we prove that
preserves the order in both directions. From the Lemma 2.5, for
, we have
Thus
is an order isomorphism.
Next to prove that
is positive homogeneous. If
,
, then for every
, we have
i.e.,
. Since
, we have
. Therefore by Lemma 2.2, there is a Jordan
-isomorphism
such that
(
) For
, we have
This completes the proof. □