Characterization of n Jordan homomorphisms on Banach algebras

In this paper we prove that every n-Jordan homomorphis varphi:mathcal {A} longrightarrowmathcal {B} from unital Banach algebras mathcal {A} into varphi -commutative Banach algebra mathcal {B} satisfiying the condition varphi (x^2)=0 Longrightarrow varphi (x)=0, xin mathcal {A}, is an n-homomorphism. In this paper we prove that every n-Jordan homomorphism varphi:mathcal {A} longrightarrowmathcal {B} from unital Banach algebras $mathcal{A} into varphi$-commutative Banach algebra $mathcal {B}. satisfiying the condition varphi (x^2)=0 Longrightarrow varphi (x)=0, xin mathcal {A}, is an n-homomorphism. This result was proved in cite {zivari1} for 3-Jordan homomorphism with the additional hypothesis that the Banach algebra mathcal {A} is unital, and it is extended for all nin mathbb {N} in cite {An}. Later, for nin lbrace 3,4 rbrace, Theorem ref {T1} proved in cite {Bodaghi1} by considering an extra condition varphi (ab^2)= varphi (b^2a), (a,bin mathcal {A}). Some significant results concerning Jordan homomorphisms and their automatic continuity on Banach algebras obtained by the author in cite {zivari}, cite {zivari2} and cite {zivari3}. In this paper, under special hypotheses we prove that every n-Jordan homomorphism varphi from Banach algebra mathcal {A} into Banach algebra mathcal {B} is an n-homomorphism.