We provide examples to shed some light on the regularity-type properties of the boundary spectrum of a Banach algebra element. In particular, we show that the boundary spectrum is generated by a set that is neither an upper nor a lower semiregularity.
"Regularity-type properties of the boundary spectrum in Banach algebras." Rocky Mountain J. Math. 49 (8) 2747 - 2754, 2019. https://doi.org/10.1216/RMJ-2019-49-8-2747