Spectrum of an element of a non unital C*-algebra

Question

I know that spectrum of an element $x$ of a unital C*-algebra is nonempty. I like to find an example of a non unital C*-algebra that has an element with empty spectrum, if it exists.

Motivation

I saw a theorem about "spectrum of an element x of a unital C*-algebra is nonempty" so I wanted to know if the unital assumption is essential.

Answer 1

Spectrum

For Banach algebras the spectrum is always nonempty.

(This is due to the Neumann series.)

For algebras the spectrum may be empty.

(Consider the algebra of polynomials.)

Extensions

For Banach algebras the spectrum depends on extensions.

(Consider the Banach algebra of the bilateral shift.)

For C*-algebras the spectrum is independent of extensions.

(This is due to the spectral permanence.)

Reference

An investigation was done in: Extensions: Spectrum