The definition of the spectrum makes sense for any algebra. I guess we can go to the unitization to make sense of it even non-unital algebras. Recalling the well-known fact that for normed algebras, the spectrum of each element is always non-empty, I was wondering if there are examples of algebras having an element with empty spectrum.
Maybe one might have some examples in mind where we actually have a topology on the algebra.
Due to the useful comments, I make my setting more precise. I only consider complex algebras and define the spectrum as $$\sigma(x) := \{\lambda\in \mathbb{C}: \lambda 1 - x \text{ not invertible}\}.$$
Take $\mathbb C(x)$ - the field of rational functions in one variable over $\mathbb C$. The elements of the algebra are of the form $\frac{p(x)}{q(x)},$ where $p,q\in\mathbb C[x]$ are polynomials with $GCD(p,q)=1$. Then the $(x-\lambda)$ is invertible for every $\lambda \in \mathbb C$. Thus $\sigma(x)$ is empty.