Generalized Multiplicative Bijections And Bijections Preserving Witness Sets Of Coexistence On Hilbert Space Effect Algebras

The topic of preservers on operator algebaras plays an important role in the theory of operator and quantum information. The Hilbert space effect algebra ε(H), that is,εH)= {T}|0≤T≤I}, where I is the identity on the Hilbert space H, and is one of central concepts of quantum measurement and quantum information. In the thesis, we devote to characterzing generalized multiplicative bijective maps and bijective maps preserving witness sets of coexistence on Hilbert space effect algebras.We obtain the following results.1. For A,B ∈ε H), W(A,B)={C ∈ε S(H)|A+B-I≤C≤A,B} is called the witness set of coexistence of A, B. It is showed that bijective maps Φ:εH)â†'ε(H) satisfies Φ(W(A, B))=W(Φ(A),Φ(B)) if and only if there exists a unitary or anti-unitary operator U on H such that Φ(A)= UAU* for every A ∈ε(H).2. With dim H≥3, assume that α,β are two positive numbers with 2α+β≠1 and Φ:ε(H)â†'ε(H) is a bijective map. We show that Φ(AαBβAα)=Φ(A)αΦ(B)βΦ(A)α holds for all A, B ∈ε(H) if and only if there exists a unitary or anti-unitary operator U on H such that Φ(A)= UAU* for every A ∈ε(H).