Non-commutativity of exponential spectrum

TL;DR

This paper demonstrates that the exponential spectrum in Banach algebras does not share the commutativity property of the standard spectrum, resolving a long-standing open problem from 1992 by linking it to topology.

Contribution

It proves that the exponential spectrum can be non-commutative, showing a fundamental difference from the classical spectrum in Banach algebras, and uses topological methods involving homotopy groups.

Findings

Exponential spectrum does not satisfy the spectrum symmetry property.

The proof relies on the non-triviality of the homotopy group π₄(GL₂(ℂ)).

The result answers a problem open since 1992.

Abstract

In a Banach algebra, the spectrum satisfies $\sigma(ab)\setminus\{0\} = \sigma(ba)\setminus\{0\}$ for each pair of elements $a,b$. We show that this is no longer true for the exponential spectrum, thereby solving a problem open since 1992. Our proof depends on the fact that the homotopy group $\pi_4(GL_2({\mathbb C}))$ is non-trivial.