Research On Maps On Double Triangle Subspace Lattice Algebras

The study of operator algebra theory began in 1930's.Non-selfadjoint operator algebra is an important branch of theory of operator algebra. Many results related to the research for maps on operator algebras have been obtained. Double triangles lattice is an important lattice in abstract lattice,and there are many good results about the study of finite rank operators on double triangles subspace lattice algebras.In this thesis,we study three important maps on double triangle lattice algebras on Banach spaces, which are centralizers,derivations and Jordan derivations.Chapter 1 introduces some basic notions and knowledge on operator algebras and double triangles subspace lattice algebras. Moreover, we state the main results of this thesis.Chapter 2 first introduces the notions of centralizer, and shows that every centralizer is quasi-spatial.Besides,we prove that local centralizers on a standard subalgebra of double triangles subspace lattice algebras are centralizers. Finally, we present a sufficient condition under which an additive map is a centralize.Chapter 3 studies derivations on double triangles subspace lattice algebras,and shows that such a derivation is quasi-spatial. Also, we present a sufficient condition under which a derivation is continual. Furthermore, it is shown that local derivations on double triangles subspace lattice algebras are derivations.Chapter 4 studies Jordan derivations on double triangles subspace lattice algebras,and shows that any Jordan derivation on a standard subalgebra of double triangles subspace lattice algebras into B ( X ) is an additive derivation.Chapter 5 summarises the main results in this thesis and points out several interesting problems which are worth studying in our future work.