Jordan triple homomorphisms on T∞(F)

Let T ∞ ( F ) be the algebra of all N × N upper triangular matrices defined over a field F of characteristic different from 2. We consider the Jordan triple homomorphisms of T ∞ ( F ) , i.e. the additive maps that satisfy the condition ϕ ( x y x ) = ϕ ( x ) ϕ ( y ) ϕ ( x ) for all x , y ∈ T ∞ ( F ) . For the case when F is a prime field we find the form of all such maps ϕ . For the general case we present the form of the surjective maps ϕ .