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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.

Let M be a von Neumann algebra and let β be a nest in M. We consider the problem of factoring an invertible element S of M as S = WA, where W is a unitary element in M and both A and A−1 are elements of M which also belong to Alg(β). It is known that this is not always possible, however, using a variant of the LU decomposition for matrices, we show that if β is an injective nest and S is an invertible operator, then there exists an isometry W such that both S−1W and W∗S belong to Alg(β). We characterize when an invertible operator factors with respect to an injective nest. We also prove a result which simultaneously generalizes results of Arveson and Gohberg and Krein. Finally, we give a number of characterizations of those nests within a factor such that every invertible operator factors with respect to the nest.