Analyticity and flows in von Neumann algebras

Let $\text{B}$ be a von Neumann algebra, let {α_t}_tεR be an ultraweakly continuous one-parameter group of ¹-automorphisms of $\text{B}$, and let $\text{U}$ be the set of all A such that for each ? in $\text{B}$₁, the function t → ?(α_t(A)) lies in H^∞($\text{R}$. Then $\text{U}$ is an ultraweakly closed subalgebra of $\text{B}$ containing the identity which is proper and non-self-adjoint if {α_t}_tεR is not trivial. In this paper, a systematic investigation into the structure theory of $\text{U}$ is begun. Two of the more note-worthy developments are these. First of all, conditions under which $\text{U}$ is a subdiagonal algebra in $\text{B}$, in the sense of Arveson, are determined. The analysis provides a common perspective from which to view a large number of hitherto unrelated algebras. Second, the invariant subspace structure of $\text{U}$ is determined and conditions under which $\text{U}$ is a reductive subalgebra of $\text{B}$ are found. These results are then used to produce examples where $\text{U}$ is a proper, non-self-adjoint, reductive subalgebra of $\text{B}$. The examples do not answer the reductive algebra question, however, because although ultraweakly closed, the subalgebras are weakly dense in $\text{B}$.