$a\in \mathcal{B}_0$ implies $a^*\in \mathcal{B}_0$. Finally, a $C^*$-subalgebra is a norm-closed $*$-subalgebra of a $C^*$-algebra. The following two results are classical [@G; @GN].

\[thm\_gelfand\] [(Gelfand)]{.nodecor} Let $\mathcal{A}_0$ be a commutative $C^*$-algebra. Then there is a compact Hausdorff space $\mathcal{X}$ such that $\mathcal{A}_0$ is isometrically $*$-isomorphic to $C(\mathcal{X})$. If $\mathcal{Y}$ is another compact Hausdorff space such that $\mathcal{A}_0$ is isometrically $*$-isomorphic to $C(\mathcal{Y})$, then $\mathcal{X}$ and $\mathcal{Y}$ are necessarily homeomorphic.

\[thm\_gn\] [(Gelfand–Naimark)]{.nodecor} Let $\mathcal{A}_0$ be a $C^*$-algebra. Then there is a complex Hilbert space $\mathcal{H}$ such that $\mathcal{A}_0$ is isometrically $*$-isomorphic to a $C^*$-subalgebra of $\mathcal{B}(\mathcal{H})$.

We will also need the following definitions and facts. An element $a\in \mathcal{A}_0$ of a $C^*$-algebra is [*unitary*]{} if $aa^*=a^*a = I$, [*Hermitian*]{} (or [*self-adjoint*]{}) if $a^* = a$, [*skew-Hermitian*]{} if $a^* = -a$, [*normal*]{} if $aa^* = a^*a$. Clearly, unitary, Hermitian and skew-Hermitian elements are all normal. Any element $a\in \mathcal{A}_0$ can be written uniquely as $a = h_1 + {\rm i}\,h_2$ with $h_1,h_2$ Hermitian and $\rm i = \sqrt{-1}$.

For any (complex) Banach algebra $\mathcal{A}_0$, the [*spectrum*]{} of an element $a \in \mathcal{A}_0$ is the set of all $\lambda\in \mathbb{C}$ such that $\lambda I - a$ is not invertible in $\mathcal{A}_0$. We denote 