On the structure of the set of algebraic elements in a Banach algebra and their liftings

TL;DR

This paper extends results on the connectedness of algebraic elements in Banach algebras, showing pathwise connectivity of certain sets and exploring distances between components, with implications for lifting theorems.

Contribution

It generalizes earlier connectedness results for algebraic elements, introduces new path constructions, and establishes bounds and lifting theorems for these elements in Banach algebras.

Findings

Connected components of algebraic sets are pathwise connected via specific paths.

Distances between connected components are bounded below by root differences.

Lifting theorems for algebraic elements are analogous to those for idempotents.

Abstract

We generalize earlier results about connected components of idempotents in Banach algebras, due to B. Sz\H{o}kefalvi Nagy, Y. Kato, S. Maeda, Z. V. Kovarik, J. Zem\'anek, J. Esterle. Let $A$ be a unital complex Banach algebra, and $p(\lambda) = \prod\limits_{i = 1}^n (\lambda - \lambda_i)$ a polynomial over $\Bbb C$, with all roots distinct. Let $E_p(A) := \{a \in A \mid p(a) = 0\}$. Then all connected components of $E_p(A)$ are pathwise connected (locally pathwise connected) via each of the following three types of paths: 1)~similarity via a finite product of exponential functions (via an exponential function); 2)~a polynomial path (a cubic polynomial path); 3)~a polygonal path (a polygonal path consisting of $n$ segments). If $A$ is a $C^*$-algebra, $\lambda_i \in \Bbb R$, let $S_p(A):= \{a\in A \mid a = a^*$, $p(a) = 0\}$. Then all connected components of $S_p(A)$ are pathwise connected (locally pathwise connected), via a path of the form $e^{-ic_mt}\dots e^{-ic_1t} ae^{ic_1t}\dots e^{ic_mt}$, where $c_i = c_i^*$, and $t \in [0, 1]$ (of the form $e^{-ict} ae^{ict}$, where $c = c^*$, and $t \in [0,1]$). For (self-adjoint) idempotents we have by these old papers that the distance of different connected components of them is at least~$1$. For $E_p(A)$, $S_p(A)$ we pose the problem if the distance of different connected components is at least $\min \bigl\{|\lambda_i - \lambda_j| \mid 1 \leq i,j \leq n, \ i \neq j\bigr\}$. For the case of $S_p(A)$, we give a positive lower bound for these distances, that depends on $\lambda_1, \dots, \lambda_n$. We show that several local and global lifting theorems for analytic families of idempotents, along analytic families of surjective Banach algebra homomorphisms, from our recent paper with B. Aupetit and M. Mbekhta, have analogues for elements of $E_p(A)$ and $S_p(A)$.