Non-commutativity of exponential spectrum

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 ${\pi }_4(G{L}_2(\mathbb{C}))$ is non-trivial.