$\mathcal{A}_0$. The definition of unitary, Hermitian, skew-Hermitian and normal matrix are the obvious ones.

It follows from the Gelfand–Naimark representation theorem (Theorem \[thm\_gn\] above) that each $A\in \mathcal{A}$ can be represented as a matrix $T_A$ of bounded linear operators, where $T_A$ acts on the direct sum $\mathscr{H} = \mathcal{H} \oplus \cdots \oplus \mathcal{H}$ of $n$ copies of a suitable complex Hilbert space $\mathcal{H}$. This fact allows us to introduce an operator norm on $\mathcal{A}$, defined as follows: $$\label{op_norm1}
%\|A\|_{op} := \sup_{\|x\|_{\mathscr{H}}=1} \|T_Ax\|_{\mathscr{H}}\,,
\|A\|:= \sup_{\|x\|_{\mathscr{H}}=1} \|T_Ax\|_{\mathscr{H}}\,,$$ where $$\|x\|_{\mathscr{H}}:=\sqrt{\|x_1\|_{\mathcal{H}}^2 + 
                             \cdots +
                             \|x_n\|_{\mathcal{H}}^2}$$ is the norm of an element $x=(x_1,\ldots ,x_n)\in \mathscr{H}$. Relative to this norm, $A\in \mathcal{A}$ is a $C^*$-algebra. Note that $\mathcal{A}$ can also be identified with the tensor product of $C^*$-algebras $\mathcal{A}_0\, \otimes \,  \mathcal{M}_n (\mathbb{C})$.

Similarly, Gelfand’s theorem (Theorem \[thm\_gelfand\] above) implies that if $\mathcal{A}_0$ is commutative, there is a compact Hausdorff space $\mathcal{X}$ such that any $A\in \mathcal{A}$ can be identified with a continuous matrix-valued function $$A: \mathcal{X} \longrightarrow \mathcal{M}_n(\mathbb{C})\,.$$ In other words, $A$ can be represented as an $n\times n$ matrix of continuous, complex-valued functions: $A = [a_{ij}(t)]$, with domain $\mathcal{X}$. The natural $C^*$-algebra norm on $\mathcal{A}$, which can be identified with $ C(\mathcal{X}) \otimes \mathcal{M}_n 