Jordan triple homomorphisms on $${\mathcal {T}}_\infty (F)$$

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