Non-commutativity of exponential spectrum

In a Banach algebra, the spectrum satisfies $σ(ab)\setminus\{0\} = σ(ba)\setminus\{0\}$ 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 $π_4(GL_2({\mathbb C}))$ is non-trivial.

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