Zero Product Preserving Maps And Applications

The study on the problem of characerizing maps on operator algebras that preserve some invariants of isomorphisms is helpful for making the structure of operator algebras to be understood better ([1]). Such invariants include some relations between operators, some sets of operators and some functions on operators, etc.. Especially, since the zero product is a very basic relation, many authors have been payed attention to the maps preserving zero products of operators. For a product operation o defined in operator algebras, a mapΦpreserves zero products (in both directions) if A(?)B=0(?)((?) )Φ(A)(?)Φ(B)=0. Surprisedly, the results on characterizing zero product preserving maps not only the research help us understander the structure of operator algebras better, but also are applied to other research fields extensively. In the present thesis, we devote to characterizing maps on several operator sets preserving zero products in both directions, and applying these results to the research of other topics, such as the general preserver problems and some problems in Quantum Mechanics. The following are the main results obtained in this thesis:1. We characterize the following kinds of maps preserving zero products in both directions. Let X and H be a Banach space and a Hilbert space on the real or complex field F respectively with dim X≥3 and dim H≥3. Let W, V(?) B(X) or B(H) be two subsets andΦ:W→V be a map satisfies A(?)B=0(?)Φ(A)(?)Φ(B)=0 for all A, B∈W. We give a characterization of the above maps for the following cases:(1) W, V (?)B(X) are subsets containing all idempotences of rank one,Φis asurjection and A(?)B=AB; (2) W, V (?)BS(H)(BS(H) is the space of self adjoint operators on H) are subsets containing all projections of rank one,Φis a surjection and A(?)B=AB; (3) W,V(?)B(H) are subsets containing all operators of rank one,Φis a bijection and A(?)B=AB(?) and A(?)B or A(?)B=AB(?)A, where J∈B(H) is a self-adjoint invertible operator, A(?)= J-1A*J; (4) W=V=CP(H),Φis a real linear surjective map and AοB=0 is replaced by A*B=AB*=0; (5) W,V (?) B(H) are subsets of positive operators containing all projections of rank one,Φis bijective and A(?)B= ATB (T is an arbitrarily given positive invertible operator).2. Applying the results in 1, we obtain characterizaions of maps on corresponding operator sets preserving numerical radius or cross norms of corresponding products of operators. 3. Applying the result 1(4) of characerizing maps preserving orthogonality in both directions on the operator sets, we give a characterizaion of distance preserving and com-pletely distance preserving maps on the Schatten-p class, and distance preserving maps on the Schatten-p class in nest algebras.4. Applying the result of classificating maps preserving generalized orthogonality in both directions on the set of positive operators contains rank one projections, we give a generalization of Wigner theorem in theory of Quantum Mechanics.5. Also applying the result of classificating maps preserving orthogonality in both directions on the set of positive operators containing rank one projections, we give a characterization of general sequential isomorphisms on Hilbert space effect algebras. Let the operation (?) be an arbitrary sequential product on the Hilbert space effect algebrasε(H) on a Hilbcrt space H with dimH≥2. If the mapΦ:ε(H)→ε(H) is bijective and satisfies thatΦ(A(?)B)=Φ(A)(?)(B) for A, B∈ε(H), then there is an either unitary or anti-unitary operator U such thatΦ(A)=UAU* for every A∈ε(H).