Infinitesimal aspects of idempotents in Banach algebras

TL;DR

This paper explores the infinitesimal structure of flag manifolds formed by idempotents in Banach algebras, introducing Stiefel bundles and connections in an infinite-dimensional setting to advance operator theory.

Contribution

It introduces and studies Stiefel bundles on flag manifolds in Banach algebras, extending classical bundle concepts to infinite dimensions and linking them with operator theory.

Findings

Defined and analyzed Stiefel bundles on flag manifolds

Connected bundle theory with operator theory in Banach algebras

Surveyed notions of connections in infinite-dimensional bundles

Abstract

We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of our investigation is the notion of connection on an infinite-dimensional bundle, and we survey some equivalent ocurrences of such a notion in the literature.