As you probably know, the classical proof of the non-emptiness of the spectrum for an element x in a general Banach algebra over C can be proven quite easily using Liouville’s theorem in complex analysis: every bounded, entire function C→C is constant.
As these two theorems seem closely related and are certainly strong and non-trivial (for instance, both of them easily imply the fundamental theorem of algebra), I wonder if it is also possible to deduce Liouville’s theorem from the non-emptiness of spectra for elements in complex Banach algebras. I guess one would like to to apply the Gelfand-Mazur theorem (which is a simple corollary of the above non-emptiness) to the Banach algebra of bounded, entire functions on C but showing that this is a division algebra is basically the same as showing that it is equal to C to begin with.
Yes, you can find such a proof in the following article:
Citation: Singh, D. (2006). The spectrum in a Banach algebra. The American Mathematical Monthly, 113(8), 756-758.
The paper is easily located behind a pay-wall (JSTOR) here.
Here is an image of the start:
As promised, the article concludes with a proof of Liouville’s Theorem following from Theorem 1 here, that is, using the non-emptiness of \sigma(a) as specified by the OP. Perhaps someone else can find a copy of this article that is freely accessible.