the spectrum of $a$ by $\sigma(a)$. For any $a\in \mathcal{A}_0$, the spectrum $\sigma(a)$ is a non-empty compact subset of $\mathbb{C}$ contained in the closed disk of radius $r=\|a\|$ centered at $0$. The complement $r(a) = \mathbb{C} \backslash \sigma(a)$ of the spectrum of an element $a$ of a $C^*$-algebra is called the [*resolvent set*]{} of $a$. The [*spectral radius*]{} of $a$ is defined as $\rho(a) = \max\{|\lambda| \,;\, \lambda\in \sigma(A)\}$. [*Gelfand’s formula*]{} for the spectral radius [@G] states that $$\label{GF}
\rho(a) = \lim_{m\to \infty} \|a^m\|^{\frac{1}{m}}\,.$$ Note that this identity contains the statement that the above limit exists.

If $a \in \mathcal{A}_0$ (a $C^*$-algebra) is Hermitian, $\sigma(a)$ is a subset of $\mathbb{R}$. If $a \in \mathcal{A}_0$ is normal (in particular, Hermitian), then $\rho(a) = \|a\|$. This implies that if $a$ is Hermitian, then either $-\|a\|\in \sigma(a)$ or $\|a\|\in \sigma(a)$. The spectrum of a skew-Hermitian element is purely imaginary, and the spectrum of a unitary element is contained in the unit circle $S^1 = \{z\in \mathbb{C}\, ; \,
 |z|=1\}$.

An element $a\in \mathcal{A}_0$ is [*nonnegative*]{} if $a=a^*$ and the spectrum of $a$ is contained in $\mathbb{R}_+$, the nonnegative real axis. Any linear combination with real nonnegative coefficients of nonnegative elements of a $C^*$-algebra 