Jordan Semi-Triple Multiplicative Maps on the Symmetric Matrices ()
1. Introduction
It is an interesting problem to study the interrelation between the multiplicative and the additive structure of a ring or an algebra. Matindale in [1] proved that every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is additive. Thus, the multiplicative structure determines the ring structure for some rings. This result was utilized by P. Šemrl in [2] to describe the form of the semigroup isomorphisms of standard operator algebras on Banach spaces. Some other results on the additivity of multiplicative maps between operator algebras can be found in [3,4]. Besides ring homomorphisms between rings, sometimes one has to consider Jordan ring homomorphisms. Note that, Jordan operator algebras have important applications in the mathematical foundations of quantum mechanics. So, it is also interesting to ask when the Jordan multiplicative structure determines the Jordan ring structure of Jordan rings or algebras.
Let 
be two rings and let 
be a map. Recall that 
is called a Jordan homomorphism if
for all
. There are two basic forms of Jordan multiplicative maps, namely1) 
(Jordan semi-triple multiplicative map) for all
2) 
(Jordan multiplicative map) for all
. It is clear that, if 
is unital and additive, then these two forms of Jordan multiplicative maps are equivalent. But in general, for a unital map, we do not know whether they are still equivalent without the additivity assumption.
The question of when a Jordan multiplicative map is additive was investigated by several authors. Let
be a bijective map on a standard operator algebra. Molnár showed in [5] that if 
satisfies
then 
is additive. Later, Molnár in [5] and then Lu in [6] considered the cases that 
preserve the operation
and
, respectively, and proved that such 
is also additive. Thus, the Jordan multiplicative structure also determines the Jordan ring structure of the standard operator algebras. Later, in [7] we proved these Jordan multiplicative maps on the space of selfadjoint operators space are Jordan ring isomorphism and thus are equivalent. In this paper, we consider the same question and give affirmative answer for the case of Jordan multiplicative maps on the Jordan algebras of all symmetric matrices. In fact, we study injective Jordan semi-triple multiplicative maps on the symmetric matrices
, and show that such maps must be additive, and hence are Jordan ring homomorphisms.
Let us recall and fix some notations in this paper. Recall that 
is called an idempotent if
. We define the order
between idempotents as follows: 
if and only if 
for any idempotents
,
. For any
, let 
be the matrix with 1 in the position 
and zeros elsewhere, and 
be the unit of
.
2. Main Results and Its Proof
In this section, we study injective Jordan semi-triple multiplicative maps on
, the following is the main result.
Theorem 2.1. An injective map
is a Jordan semi-triple multiplicative map, that is
(2.1)
if and only if there is an injective homomorphism 
of 
and a complex orthogonal matrix 
such that
for all
.
Firstly, we give some properties of injective Jordan semi-triple multiplicative maps on
.
Lemma 2.2. Let 
be an injective Jordan semi-triple multiplicative map. Then 
sends idempotents to tripotents and moreover1) 
is an idempotent and
for all
, in particular
2) 
commutes with 
for every
;
3) 
is an idempotent for each idempotent
;
4) A map 
defined by
for all
, is a Jordan semi-triple multiplicative map, which is injective if and only if 
is injective.
For 
defined in Lemma 2.2, we can see that
and 
for any idempotents
. Therefore, we have Corollary 2.3. Let 
and
be an injective Jordan semi-triple multiplicative map. Then
. In the case
, for each idempotent 
the rank of 
is equal to the rank of
. In particular,
and
Now we give proof of Theorem 2.1. The main idea is to use the induction on
, the dimension of the matrix algebra, after proving the result for 
matrices.
Proof of Theorem 2.1. In order to prove Theorem 2.1, it suffices to characterize
. Note if
then
that is 
is invertible and
By Lemma 2.1, 
commutes with 
for all
. It follows that 
commutes with 
for all
. Therefore, if
, 
must be a scalar matrix. As 
and hence
has the desired form.
Therefore, we mainly characterize
. The proofs are given in two steps.
Step 1. The proof for
.
The matrix 
is an idempotent of rank one. By Corollary 2.3, 
is a rank one idempotent. It is well known that every idempotent matrix in 
can be diagonalizable by complex orthogonal matrix. Thus, there exists a 
orthogonal matrix 
such that
Without loss of generality, we may assume that
By Corollary 2.3 and from the following fact
and
we conclude that
or
Let
, by replacing 
with 
if necessary, we may assume that
.
For
, since 
is a rank one idempotent and satisfying 
and
we have
. Now for any
let
. Then
Thus, the
entry of 
depends on the 
entry of
only. Therefore, there exist injective functionals
such that 
satisfy respectively 
and
and
.
From
, it is easy to verify that 
is multiplicative. Next we prove that
. Let
since 
A and
, we have
and
, hence 
or
with
.
Thus, 
and 
since
is multiplicative. Let
, then
. Note that 
and
that is
This implies 
and
. Now by the fact 
and
, we get
. For any
, since
thus
.
Next we prove that 
is additive. Since
and thus we have
for any
. Moreover by the fact 
one can get that
and
.
Finally, we prove
for any
. Let
.
By the fact that 
and
we get 
and 
for any
.
Step 2. The induction.
Let
then 
is a rank 
idempotent, so is 
by Corollary 2.3. Therefore, there exists a orthogonal matrix 
such that
. Replacing 
by the map 
we may assume that
.
For any 
let
. Then 
implies
It follows that 
for some matrix
. Define the map 
on 
by
. It is easy to check that 
is an injective Jordan semi-triple multiplicative map on
. Furthermore, 
implies that
. By the induction hypothesis there is a 
orthogonal matrix 
and an injective homomorphism 
on 
such that 
Let 
be the matrix
. Without loss of generality, we assume that 
for all
. This is equivalent to
. For any
with 
and
, we have
.
Thus,
(*)
Let us define matrices 
for each 
by
For an arbitrary
, From (*) we have
Then there exists 
and 
such that
From the equality 
we get that 
and
. These equality implies that 
and
Hence only the 
entries 
of 
are nonzero and
. It follows that
Next, take any two distinct
. From
and using (*) , we get
which implies that
. Let
, then
, so we may assume that
. Furthermore by the equality
and
, we obtain 
Next we prove that 
for any
.
Let us fix some
. As
, there is another 
such that
Then for any
,
and
.
Thus, for any
where 
has only one nonzero entry in the 
position, we have
. For any
, let
and
.
From
, we have
And
. For any
, since
where 
and 
have only one nonzero entry 
and 
in the 
and 
position respectively, 
is equal to the 
entry of
, thus we have
and so
. The proofs are complete.
By Theorem 2.1, we can characterize another two forms of Jordan multiplicative maps on
.
Theorem 2.4. An injective map
satisfies
(2.2)
if and only if there is an injective homomorphism 
on 
and a complex orthogonal matrix 
such that
for all
.
Proof. Let
in Equation (2.2), we get
that is, 
is a Jordan semi-triple multiplicative map. Consequently, 
has the desired form by Theorem 2.1.
Since every ring homomorphism on
is an identity map, thus by Theorem 2.1, Theorem 2.4, we get Corollary 2.5. Let 
be an injective map. Then the following condition are equivalent1) 
2) 
3) there is a real orthogonal matrix
such that
for all
.
At the end of this section, we characterize bijective maps on 
preserving
.
Theorem 2.6. A bijective map 
satisfies
(2.3)
if and only if there is a ring isomorphism 
on 
and a complex orthogonal matrix
such that
for all 
Proof. It is enough to check the “only if” part. Letting
in Equation (2.3), we get
Taking 
and
, we get 
and thus
(2.4)
Letting 
in Equation (2.3), we get
.
Taking
, we get
.
Multiplying this equality by 
from the left side, by Equation (2.4) we get
for any
, and hence 
for some scalar
. By Equation (2.4), we obtain 
If
, let
, then 
also meets Equation (2.3) and
. So without loss of generality, we assume
. By letting 
and
in Equation (2.3), we get
and 
for all
. Consequently
Now letting
in Equation (2.3) we get
.
Thus, 
and 
by taking
in Equation (2.3). Therefore, 
has desired form by surjectivity of 
and Theorem 2.1.
In particular, we have Corollary 2.7. A bijective map 
satisfies
if and only if there is a real orthogonal matrix
such that
for all
.
Remark 2.8. We do not know whether the surjective assumption in Theorem 2.6 and Corollary 2.7 can be omitted.
NOTES