Centralizers, (α,β)-derivations And Anti-isomorphisms On J-subspace Lattice Algebras

Non-selfadjoint and reflexive operator algebra is an important branch of the theory of operator algebra. Since J. Ringrose began to study nest algebras in the 1960s, many people have devoted themselves to the study of non-selfadjoint and reflexive operator algebras including nest algebras, commutative subspace lattice algebras, completely distributive subspace lattice algebras and so on, and obtain a lot of beautiful achievements. J -subspace lattice algebra is an important non-selfadjoint and reflexive operator algebra, and is entirely different from nest algebra, because the lattice of nest algebras is totally ordered by inclusion, but arbitrary intersection of the elements in J -subspace lattices is a null space.In this thesis, we study three important maps of J -subspace lattice algebras on Banach spaces, which are centralizers, (α,β)-derivations and anti-isomorphisms. The general forms of these maps are described.Chapter 1 introduces the basic notions and properties of J -subspace lattice algebras and the three maps mentioned above. Also, we state the main results of this thesis.Chapter 2 studies centralizers on J -subspace lattice algebras, and shows that such a centralizer is quasi-spatial implemented.Chapter 3 studies local centralizers on J -subspace lattice algebras . We prove every linear and local centralizer on J -subspace lattice algebras is a centralizer. Also, let F ( L ) be the algebra of all finite rank operators in a J -subspace lattice algebra . It is proved that every linear mapΦon F ( L ) such thatΦ( P ) =Φ( P )P = PΦ( P) for each idempotent P in F ( L ) must be a centralizer.Chapter 4 studies (α,β)- derivations on J -subspace lattice algebras, and describes the form of (α,β)-derivations.Chapter 5 characterizes anti-isomorphisms on J -subspace lattice algebras, and the result says that such an anti-isomorphism is qusi-spatial implemented.