Commutativity and orthogonality of similarity orbits in Banach algebras
For a semisimple unital Banach algebra $A$ over $\mathbb{C}$, and elements $a,b\in A,$ we show that the similarity orbits, $\mathrm{orb}(a)$ and $\mathrm{orb}(b)$, over the principal component of the invertible group of $A$ commute precisely when there is at least one nonzero complex number not belonging to the spectrum of any product $a^\prime b^\prime$ -- where $(a^\prime,b^\prime)\in\mathrm{orb}(a)\times\mathrm{orb}(b)$. In this case, the polynomially convex hull of the spectra of the $a^\prime b^\prime$ is constant. When $\mathrm{orb}(a)=\mathrm{orb}(b)$, then $a$ is central under the aforementioned assumption -- and the result then generalizes part of an old theorem due to J. Zem\'anek. We show further that the two classical characterizations of commutative Banach algebras via the spectral radius can be algebraically localized in the sense of `local' implies `global'. Thereafter, in Section 3, we give a (somewhat weaker) localization of the above situation involving spectral perturbation on small neighborhoods in a similarity orbit. Finally, we apply the above results to algebraic elements and idempotents in particular, so that orthogonality of similarity orbits of two idempotents is equivalent to a pair of spectral radius properties. To conclude with, a couple of localization theorems specific to idempotents and algebraic elements are presented. Similar statements to all of the above hold if $ a^\prime b^\prime $ is replaced by $ a^\prime + b^\prime $, $ a^\prime - b^\prime $, or $ a^\prime + b^\prime-a^\prime b^\prime $.