Local and global liftings of analytic families of idempotents in Banach algebras

TL;DR

This paper develops local and global lifting theorems for analytic families of idempotents in Banach algebras, under spectral conditions, extending previous results with new spectral and topological techniques.

Contribution

It introduces new local and global lifting theorems for analytic idempotent families in Banach algebras, including cases with disconnected spectra and self-adjoint idempotents.

Findings

Local lifting theorem for spectra not disconnecting a2a2

Global lifting theorem for spectra equal to a2a2

Mutually orthogonal families of idempotents can be lifted

Abstract

Generalizing results of our earlier paper, we investigate the following question. Let $\pi(\lambda) : A \to B$ be an analytic family of surjective homomorphisms between two Banach algebras, and $q(\lambda)$ an analytic family of idempotents in $B$. We want to find an analytic family $p(\lambda)$ of idempotents in $A$, lifting $q(\lambda)$, i.e., such that $\pi(\lambda)p(\lambda) = q(\lambda)$, under hypotheses of the type that the elements of $\text{Ker}\, \pi(\lambda)$ have small spectra. For spectra which do not disconnect $\Bbb C$ we obtain a local lifting theorem. For real analytic families of surjective $^*$-homomorphisms (for continuous involutions) and self-adjoint idempotents we obtain a local lifting theorem, for totally disconnected spectra. We obtain a global lifting theorem if the spectra of the elements in $\text{Ker}\, \pi(\lambda)$ are $\{0\}$, both in the analytic case, and, for $^*$-algebras (with continuous involutions) and self-adjoint idempotents, in the real analytic case. Here even an at most countably infinite set of mutually orthogonal analytic families of idempotents can be lifted to mutually orthogonal analytic families of idempotents. In the proofs, spectral theory is combined with complex analysis and general topology, and even a connection with potential theory is mentioned.