Jordan $-$homomorphisms between unital $C^-$algebras

Abstract: Let

A, B

be two unital

C^{*} -

algebras. We prove that every almost unital almost linear mapping

h : A l o n g r i g h t a r r o w B

which satisfies

h (3^{n} u y + 3^{n} y u) = h (3^{n} u) h (y) + h (y) h (3^{n} u)

for all

u i n U (A)

, all

y i n A

, and all

n = 0, 1, 2, . . .

, is a Jordan homomorphism. Also, for a unital

C^{*} -

algebra

A

of real rank zero, every almost unital almost linear continuous mapping

h : A l o n g r i g h t a r r o w B

is a Jordan homomorphism when

h (3^{n} u y + 3^{n} y u) = h (3^{n} u) h (y) + h (y) h (3^{n} u)

holds for all

u i n I_{1} (A_{s a})

, all

y i n A,

and all

n = 0, 1, 2, . . .

. Furthermore, we investigate the Hyers--Ulam--Rassias stability of Jordan

* -

homomorphisms between unital

C^{*} -

algebras by using the fixed points methods.

Jordan $*-$homomorphisms between unital $C^*-$algebras