For every $N \in \mathbb{N}$, let $M_N(\mathbb{C})$ be the unital complex algebra of $N \times N$ complex matrices with unit $I_N$ and ordinary matrix multiplication. Let

\begin{align*} *: M_N(\mathbb{C}) \to M_N(\mathbb{C}) \end{align*}

be the involution defined by sending each matrix $A \in M_N(\mathbb{C})$ to its conjugate transpose $A^* \in M_N(\mathbb{C})$. Define the normalized trace

\begin{align*} \operatorname{tr}_N: M_N(\mathbb{C}) \to \mathbb{C}, \quad A \mapsto \frac{1}{N}\sum_{i=1}^{N} A_{ii}. \end{align*}

Then $(M_N(\mathbb{C}), \operatorname{tr}_N)$ is a faithful tracial $*$-probability space: $M_N(\mathbb{C})$ is a unital complex $*$-algebra, $\operatorname{tr}_N$ is complex-linear, $\operatorname{tr}_N(I_N)=1$, $\operatorname{tr}_N(A^*A) \ge 0$ for every $A \in M_N(\mathbb{C})$, $\operatorname{tr}_N(A^*A)=0$ implies $A=0$, and $\operatorname{tr}_N(AB)=\operatorname{tr}_N(BA)$ for every $A,B \in M_N(\mathbb{C})$.