Jordan Triple Maps On The Spin Factor And Triples On The Jordan Algebra Mapping

Recently many people have investigated Jordan maps in oprator algebras vastly. And the additivity of Jordan triple maps is quite valuable to be considered too. Let R be the field of real numbers and H be a real Hilbert space of dimension at least 2. Let A= H+R·1 be the spin factor corresponding to H. In this note, we prove that if a bijective mapφfrom A to itself satisfiesφ({abc})={φ{a)φ(b)φ{c)} for all a,b,c∈A, andφ|R., is additive, then there is a unique unitary operator U on H, such thatφ(x+a·1)= Ux+α·1 orφ(x+a·1)=-Ux-α·1 for every x∈H,α∈R.Let A and B be Jordan algebra. The bijectionφ:Aâ†'B is called a Jordan triple map, ifφ({abc})={φ(a)φ(b)φ(c)} for all a,b,c∈A. If A contains a non-trivial idempotent p, and the Peirce decomposition A= A1(?)A1/2(?)A0 of A with respect to p, and satisfies that (1)αi∈Ai(i= 1,0), ifαi (?)t1/2= 0 for all t1/2∈A1/2, thenαi= 0, every Jordan triple map from A onto B is additive.