Spin因子上的Jordan三元映射

Jordan Triple Maps on Spin Factors

设R是实数域,H是维数≥2的实的Hilbert空间并且A=H+R.1为对应于的Spin因子.如果从A到它自身的双射Ф满足:(1)任给a,b,c∈A,都有Ф({abc})={Ф(a)Ф(b)Ф(c)};(2)Ф|R.1是可加的,则H上存在唯一的酉元U,使得任给x∈H,α∈R,都有Ф(x+α.1)=Ux+α.1或Ф(x+α.1)=-Ux-α.1.

Let R be the field of real numbers, H be a real Hilbert space of dimension at least 2 and A=H＋R·1 be the spin factor corresponding to H. The existence and uniqueness of a unitary operator U on H are proved, such that Ф（x＋α·1）=Ux＋α·1 or Ф（x＋α·1）=Ux-α·1 for every x∈H,α∈R,, if a bijective map Ф from A to itself satisfies Ф（{abc}）={Ф（a）Ф（b）Ф（c）}; for all a, b, c ∈ A, and Ф｜R·1 is additive.