Some applications of direct integral decompositions of $W^{\ast }$-algebras

Abstract:

Let $\mathcal {A}$ be a ${W^{\ast }}$-algebra and let $A \in \mathcal {A}$. $\mathcal {K}(\mathcal {A})$ and $C(A)$ represent certain convex subsets of $\mathcal {A}$. We prove the following via direct integral theory: (1) If $\mathcal {A}$ is of type ${{\text {I}}_\infty }$, ${\text {II}}_\infty$, or III, then $C(A) = \{ 0\}$ iff ${\text {A}} \in \mathcal {K}(\mathcal {A})$. (2) If $\mathcal {A}$ is of type I or II, then $\mathcal {K}(\mathcal {A})$ is strongly dense in $\mathcal {A}$. (3) If $\mathcal {A}$ is of type ${{\text {I}}_\infty }$, ${\text {II}}_\infty$, or III and $\mathcal {B}$ is a ${W^{\ast }}$-subalgebra of $\mathcal {A}$, we give sufficient conditions for a Schwartz map $P$ of $\mathcal {A}$ into $\mathcal {B}$ to annihilate $\mathcal {K}(\mathcal {A})$. Several preliminary lemmas that are useful for direct integral theory are also proved.