is nonnegative; in other words, the set of all nonnegative elements in a $C^*$-algebra $\mathcal{A}_0$ form a (nonnegative) [*cone*]{} in $\mathcal{A}_0$. For any $a\in \mathcal{A}_0$, $aa^*$ is nonnegative, and $I + aa^*$ is invertible in $\mathcal{A}_0$. Furthermore, $\|a\| = \sqrt{\rho(a^*a)} = 
\sqrt{\rho(aa^*)}$, for any $a\in \mathcal{A_0}$.

Finally, we note that if $\|\cdot\|_*$ and $\|\cdot\|_{**}$ are two norms with respect to which $\mathcal{A}_0$ is a $C^*$-algebra, then $\|\cdot\|_* = \|\cdot\|_{**}$.

Matrices over a $C^*$-algebra
=============================

Let $\mathcal{A}_0$ be a $C^*$-algebra. Given a positive integer $n$, let $\mathcal{A} = \mathcal{M}_n(\mathcal{A}_0)$ be the set of $n\times n$ matrices with entries in $\mathcal{A}_0$. Observe that $\mathcal{A}$ has a natural $C^*$-algebra structure, with matrix addition and multiplication defined in the usual way (in terms, of course, of the corresponding operations on $\mathcal{A}_0$). The involution is naturally defined as follows: given a matrix $A=[a_{ij}] \in \mathcal{A}$, the adjoint of $A$ is given by $A^* = [a_{ji}^*]$. The algebra $\mathcal{A}$ is obviously unital, with unit $$I_n = \left[ \begin{array}{cccc}
               I & 0 &\ldots  & 0 \\ 0 & \ddots & \ddots & \vdots \\
               \vdots  & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & I
             \end{array}
      \right]$$ where $I$ is the unit of 