Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra

TL;DR

This paper constructs smooth connecting paths within connected components of algebraic elements in Banach algebras, extending previous results about idempotents and self-adjoint elements, and explores their geometric properties.

Contribution

It introduces new methods to connect algebraic elements satisfying polynomial equations in Banach algebras, generalizing earlier results and providing geometric insights.

Findings

Every non-central element lies on a complex line satisfying the polynomial equation.

Constructs of connecting paths in connected components of algebraic elements.

Formulation of open questions for future research.

Abstract

Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a $C^*$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we will prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions.