Factorization in operator algebras

Abstract:

Let $M \subseteq B(H)$ be a $\sigma$-finite von Neumann algebra and let $\mathcal {L}$ be a lattice of projections associated with a unitary representation $\left \{ {{W_t}} \right \}$ of a compact group with an ordered dual. ($\mathcal {L}$ is not necessarily contained in $M$.) Assume that $M$ is invariant under ad ${W_t}$ for every $t$. Then, whenever $T$ is an invertible operator in $M$ that can be factored as $T = UA$ where $U$ is a unitary operator in $B(H)$ and $A$ lies in $\operatorname {alg} \mathcal {L} \cap {(\operatorname {alg} \mathcal {L}{\text {)}}^{ - 1}}$, then $U$ and $A$ can be chosen in $M$. As corollaries we derive results about factorization with respect to CSL subalgebras and analytic subalgebras of von Neumann algebras.