Idempotents in nest algebras

Abstract

The algebraic equivalence and similarity classes of idempotents within a nest algebra Alg β are completely characterized. We obtain necessary and sufficient conditions for two idempotents to be equivalent or similar. Our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a “dimension function” from β × β into N ∪ { ∞ }. We complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents. Also, to each sequence of dimension functions, there corresponds a commuting sequence of idempotents. A criterion is obtained for when an idempotent is similar to a subidempotent of another. The mapping which sends an equivalence class (or idempotent) to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a Type I W ∗-algebra to its center valued trace. We prove that almost commuting, similar idempotents are homotopic; this contrasts with the situation in certain C ∗-algebras. Using this, we show that similar, simultaneously diagonalizable idempotents are homotopic, and in the continuous nest case, every diagonal idempotent is homotopic to a core projection.