Let ,4 and B be unital C＊-algebras, and let J∈A,L∈B be Hermitian invertible elements. For every T∈A and S∈B, define T^＋J=J^-1T＊J and ,S^＋L=^L-1,S＊L. Then in such a way we endow the C＊-algebras A and B with indefinite structures. We characterize firstly the Jordan （J, L）＋homomorphisms on C＊-algebras. As applications, we further classify the bounded linear maps Ф： A →B preserving （J, L）-unitary elements. When ,4 = B（H） and B=B（K）, where H and K are infinite dimensional and complete indefinite inner product spaces on real or complex fields, we prove that indefinite-unitary preserving bounded linear surjections are of the form T→UVTV^-1 （任意T∈B（H）） or T→UVT^＋V^-1（任意T∈B（H））, where U∈B（K） is indefinite unitary and, V ： H→ K is generalized indefinite unitary in the first form and generalized indefinite anti-unitary in the second one. Some results on indefinite orthogonality preserving additive maps are also given.
Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempo-tents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ {-1,1,2,3,1/2,1/3} and A, B ∈ B(H), A - λB ∈ I(H) (?) Φ(A) - λΦ(B) ∈ I(H), then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT-1 for all A ∈ B(H), or Φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A-iB ∈ I(H) (?) Φ(A) -ιΦ(B) ∈ I(H), here ι is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.
CUI Jianlian & HOU Jinchuan Department of Mathematical Science, Tsinghua University, Beijing 100084, China
The authors extend Hua’s fundamental theorem of the geometry of Hermitian matri- ces to the in?nite-dimensional case. An application to characterizing the corresponding Jordan ring automorphism is also presented.
Let H and K be indefinite inner product spaces. This paper shows that a bijective map φ：B（H） →B（K） satisfies φ（AB^＋ ＋ B^＋A） = φ（A）φ（B）^＋ ＋ φ（B）^＋φ（A） for every pair A, B ∈ B（H） if and only if either φ（A） = cUAU^＋ for all A or φ（A） = cUA^＋U^＋ for all A; φ satisfies φ（AB^＋A） = φ;（A）φ;（B）^＋φ;（A） for every pair A, B ∈ B（H） if and only if either φ（A） = UAV for all A or φ（A） = UA^＋V for all A, where At denotes the indefinite conjugate of A, U and V are bounded invertible linear or conjugate linear operators with U^tU = c^-1I and V^＋V = cI for some nonzero real number c.
