Analytic operator algebras (factorization and an expectation)

Abstract:

Let $M$ be a $\sigma$-finite von Neumann algebra and ${\{ {\alpha _t}\} _{t \in T}}$ a periodic flow on $M$. The algebra of analytic operators in $M$ is $\{ a \in M:{\text {sp}_\alpha }(a) \subseteq {{\mathbf {Z}}_ + }\}$ and is denoted ${H^\infty }(\alpha )$. We prove that every invertible operator $a \in {H^\infty }(\alpha )$ can be written as $a = ub$, where $u$ is unitary in $M$ and $b \in {H^\infty }(\alpha ) \cap {H^\infty }{(\alpha )^{ - 1}}$. We also prove inner-outer factorization results for $a \in {H^\infty }(\alpha )$. Another result represents ${H^\infty }(\alpha )$ as the image of a certain nest subalgebra (of a von Neumann algebra that contains $M$) via a conditional expectation. As corollaries we prove a distance formula and an interpolation result for the case where $M$ is an injective von Neumann algebra.