Reflexivity, elementary operators and cohomology
Let H be a separable complex Hilbert space and let B(H) be the set of all bounded operators on H. In this dissertation, we show that if S is a n-dimensional subspace of B( H), then S is [ 2n ]-reflexive, where [t] denotes the largest integer that is less than or equal to t. We obtain some lattice-theoretic conditions on a subspace lattice L which imply alg L , is strongly rank decomposable. Let S be either a reflexive subspace or a bimodule of a reflexive algebra. We find some conditions such that T has a rank one summand in S and S has strong rank decomposability. Let S ( L ) be the set of all operators on H that annihilate all the operators of rank at most one in alg L . Katavolos, Katsoulis and Longstaff show that if L is a subspace lattice generated by two atoms, then S ( L ) is strongly rank decomposable. They ask whether S ( L ) is strongly rank decomposable if L is an atomic Boolean subspace latttice with more than two atoms. For any n &ge; 3, we construct an atomic Boolean subspace lattice L on H with n atoms such that there is a finite rank operator T in S ( L ) such that T does not have a rank one summand in S ( L ). This answers their question negatively. We also discuss isomorphisms of reflexive algebras. We introduce a new concept called "bounded reflexivity" for a subspace of operators on a normed space. We explore the properties of bounded reflexivity, and we compare the similarities and differences between bounded reflexivity and the usual reflexivity for a subspace of operators. We discuss the relations of bounded reflexivity of subspaces of B( H) and complete positivity of elementary operators on B( H). As applications of bounded reflexivity, we give shorter proofs of some well known results about positivity and complete positivity of elementary operators. By using those ideas, we study properties of a C*-algebra in which every n-positive elementary operator is completely positive. We study the derivations in nonselfadjoint algebras. We research derivations on a nest subalgebra of von Neumann algebras. We also consider two cohomology theories, the norm continuous cohomology and the normal cohomology on some nonselfadjoint algebras. Those algebras contain reflexive algebras whose invariant subspace lattices are tensor products of nests and reflexive algebras whose invariant subspace lattices are generated by two atoms. We obtain for those algebras A that Hnc ( A , B (H)) = Hnw ( A , B(H)).
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