Analytic bundle structure on the idempotent manifold.

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and I (X) be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from L (X) , the subset I (X) is a locally trivial analytic affine-Banach bundle over the Grassmann manifold G (X) , via the map κ that sends Q ∈ I (X) to Q(X), such that the affine-Banach space structure on each fiber is the one induced from L (X) . Using this, we show that if H is a real or complex Hilbert space, then the assignment (E , T) ↦ T ∗ ∘ P E ⊥ + P E , where E ∈ G (H) and T ∈ L (E , E ⊥) , induces a real bi-analytic bijection from the total space of the tangent bundle, T (G (H)) , of G (H) onto I (H) (here, E ⊥ is the orthogonal complement of E, P E ∈ L (H) is the orthogonal projection onto E, and T ∗ is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every E ∈ G (H) , we identify L (E , E ⊥) with a subspace of L (H) via the embedding S ↦ S ∘ P E , then the inclusion map from T (G (H)) to the trivial Banach bundle G (H) × L (H) is a real analytic immersion. Through this, we give a concrete idempotent in M n 2 (C (G (K n))) that represents the K-theory class of the tangent bundle T (G (K n)) , when K is either the real field or the complex field. [ABSTRACT FROM AUTHOR]