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As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in complex analysis: every bounded, entire function $\mathbb{C} \to \mathbb{C}$ 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 $\mathbb{C}$ but showing that this is a division algebra is basically the same as showing that it is equal to $\mathbb{C}$ to begin with.

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    A proof of nonempty spectra => fundamental theorem of algebra can go as follows, and applies to any normed field $(K,\lVert\cdot\rVert)$. Let $A$ be the algebraic closure of the completion $\hat K$ of $K$. It is standard that there is a unique extension of the norm from $\hat K$ to $A$. Then, for any $a\in A$, let $\lambda$ be in its spectrum (as a $K$-algebra). As $a-\lambda$ is not invertible, we have $a=\lambda$, so $K=A$ is algebraically closed,2013-08-15

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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:

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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.