Exponential-spectral commutativity on classical Banach spaces

Determine whether exponential-spectral commutativity can fail in the algebra of bounded operators on a classical sequence or function space, and characterize the structural properties of a Banach space that rule out such a failure.

Background

The paper constructs a failure of exponential-spectral commutativity on a specially constructed Bourgain–Delbaen space and explains that stable rank one rules out such failures. It leaves unresolved whether analogous failures occur for more familiar classical sequence or function spaces, as well as which general structural features of a Banach space prevent them.