Some applications of direct integral decompositions of 𝑊*-algebras

Abstract

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