Jordan Semi-Triple Multiplicative Maps on the Symmetric Matrices *

In this paper, we show that if an injective map  on symmetric matrices   n S C satisfies then           , , n ABA A B A A B S        C ,   Φ t f A SA S   for all   n A S C  , where f is an injective homomorphism on , is a complex orthogonal matrix and C S f A is the image of A under f applied entrywise.


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 , A B R  .There are two basic forms of Jordan multiplicative maps, namely, 1)  (Jordan multiplicative map) for all , A B R  .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]  , 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 PQ for any idempotents , .For any 1 , , let

Main Results and Its Proof
In this section, we study injective Jordan semi-triple multiplicative maps on , the following is the main result.

 
is a Jordan semi-triple multiplicative map, that is if and only if there is an injective homomorphism f of and a complex orthogonal matrix such that Firstly, we give some properties of injective Jordan semi-triple multiplicative maps on .

 
for all , is a Jordan semi-triple multiplicative map, which is injective if and only if is injective.
Now we give proof of Theorem 2.1.The main idea is to us o pro 2.1, it e the induction on n , the dimension of the matrix algebra, after proving t result for 2 2  matrices.
Proof of Theorem 2.1.In order t ve Theorem he suffices to characterize  .Note if and hence  has the de , we mainly haracterize 0 0 we conclude that or Let , by replacing  with For 22 E , since Thus, the entry of depends on the ent

 
, th i i ry of A only.Th e, there exist injective als s erefor uch that function , ,: 12 21 and   . Now by the fact AJA J  and Next we prove that f is additive.Since and thus we have It follow for some matrix .Define the map on s that Let us define matrices for each by or an arbitrary , From (*) we have Then there exists and using (*) , we get , we obtain Next we prove that Thus, for any where x  C position, we have has only one nonzero entry in the th i From , we have And

By Theorem
The proofs are complete.
2.1, we can characterize a her two forms of Jordan multiplicative maps on

   
: At the end of this section, we characterize bijective maps on (2.3) if and only if there is a ring isomorphism f on and a complex orthogonal matrix such that Taking 2 Multiplying this equality by 2