Research On Double Triangle Subspace Lattice Algebras And Maps On These Algebras

The study of operator algebra theory began in 1930's. With fast development of operator algebra theory, it has become a hot branch in modern mathematics. In order to discuss structure of operator algebra, many scholars have studied some types of linear maps on operator algebra, among which spatial and quasi-spatial problems on algebraic isomorphisms and derivations of operator algebra drew many scholars' attention. Based on the results, strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space was studied. When studying operator algebra, researchers consider the condition under which some properties of linear maps on operator algebra be determined by local properties of them. We research local derivations, linear maps derivable at zero point of strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space and localφ-derivations on some CSL algebras. Furthermore, we study an elementary operator and a Jordan isomorphism of strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space. Finally, we investigate the reflexivity of six elements subspace lattices on a non-zero complex separable Hilbert space.This thesis includes five chapters.Chapter One introduces some basic knowledge about operator algebras and operator theory, especially some notations and development about non-self-adjoint operator algebras. Furthermore, some basic knowledge and important conclusion about double triangle subspace lattice algebra were presented.Chapter Two studies algebraic isomorphisms, derivations, linear maps derivable at zero point and local derivations of strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space. Firstly, properties of strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space were studied. It was demonstrated that every operator in it is a non-zero single element if and only if it is a rank two operator. Secondly, it was proved that every algebraic isomorphism and derivation between two strong double triangle subspace lattice algebra are quasi-spatial. Thirdly, the linear map derivable at zero point of strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space was characterized. Finally, it was shown that every local derivation from strong double triangle subspace lattice algebra into B(X) is a derivation.Chapter Three investigates abstract elementary operators and Jordan isomor-phisms of strong double triangle subspace lattice algebra on a non-zero complex reflexive Banach space. Firstly, properties of a surjective elementary operator of strong double triangle subspace lattice algebra on a non-zero complex reflexive Ba-nach space were studied. It was demonstrated that every surjective elementary operator preserves rank two operators. Secondly, the surjective elementary operator of strong double triangle subspace lattice algebra was characterized. Properties of idempotent operators on strong double triangle subspace lattice algebra were in-vestigated. Finally, it was shown that every Jordan isomorphism on strong double triangle subspace lattice algebra preserves rank two operators.Chapter Four studies localφ-derivations on some CSL algebras on a complex separable Hilbert space. The lattice generated by finite commuting independent nests and completely distributive and commutative nests are two special types of CSL. Firstly, it was proved that every norm continuous localφ-derivation on a FCIN algebra is aφ-derivation. Secondly, it was shown that every norm continuous localφ-derivation on a CDC algebra is aφ-derivation.Chapter Five investigates the reflexivity of six elements subspace lattices on a complex separable Hilbert space H It was concluded that the six elements subspace lattices is reflexive if a subspace lattice (?) with six elements on (?) is isomorphic to one of (1), (2), (3) and (9) in Figure 1. If the subspace lattice (?) on a finite dimensional Hilbert space is isomorphic to one of (6), (7) and (10) in Figure 1, it is not reflexive. If the subspace lattice (?) realized by difference one dimension is isomorphic to one of (4), (5) and (8) in Figure 1, it is reflexive.