Flows and Homotopies of Banach Lie algebroids

Abstract: We establish flow and homotopy tools for Banach Lie algebroids that do not rely on compactly supported extensions. Under suitable hypotheses, we construct parameter-dependent local extensions of sections along algebroid paths, complete lifts and their fibrewise-linear evolutions, and a Bochner-integral variation-of-constants formula. We then prove that an admissible-path homotopy is equivalently a Lie algebroid morphism from the parameter square and a solution of the associated transport equation. This provides the extension, transport and infinitesimal variation theory required for a local integration program of Banach Lie algebroids to be developed in a sequel.