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.