Analytic bundle structure on the idempotent manifold

Abstract

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and $$\mathcal {I}(X)$$ 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 $$\mathcal {L}(X)$$ , the subset $$\mathcal {I}(X)$$ is a locally trivial analytic affine-Banach bundle over the Grassmann manifold $$\mathscr {G}(X)$$ , via the map $$\kappa $$ that sends $$Q\in \mathcal {I}(X)$$ to Q(X), such that the affine-Banach space structure on each fiber is the one induced from $$\mathcal {L}(X)$$ . Using this, we show that if H is a real or complex Hilbert space, then the assignment $$\begin{aligned} (E,T)\mapsto T^*\circ P_{E^\bot } + P_{E}, \quad \text {where}\quad E\in \mathscr {G}(H)\quad \text {and}\quad T\in \mathcal {L}(E,E^\bot ), \end{aligned}$$ induces a real bi-analytic bijection from the total space of the tangent bundle, $$\mathbf {T}(\mathscr {G}(H))$$ , of $$\mathscr {G}(H)$$ onto $$\mathcal {I}(H)$$ (here, $$E^\bot $$ is the orthogonal complement of E, $$P_E\in \mathcal {L}(H)$$ 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 $$E\in \mathscr {G}(H)$$ , we identify $$\mathcal {L}(E,E^\bot )$$ with a subspace of $$\mathcal {L}(H)$$ via the embedding $$S\mapsto S\circ P_E$$ , then the inclusion map from $$\mathbf {T}(\mathscr {G}(H))$$ to the trivial Banach bundle $$\mathscr {G}(H)\times \mathcal {L}(H)$$ is a real analytic immersion. Through this, we give a concrete idempotent in $$M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )$$ that represents the K-theory class of the tangent bundle $$\mathbf {T}(\mathscr {G}(\mathbb {K}^n))$$ , when $$\mathbb {K}$$ is either the real field or the complex field.