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 ≥ 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))

Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra $A_n$. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a $*$-extendible representation $\sigma$. A $*$-extendible representation of $A_n$ is ``regular'' if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given $*$-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of $\sigma(A_n)$ is weak-$*$ dense in the unit ball of the associated free semigroup algebra if and only if $\sigma$ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.Comment: 26 pages, prepared with LATeX2e, submitted to Journal of Functional Analysi