Proposition 8

If $g$ is a linear form on the $C$-module $\mathbf{M}_n(C)$ such that $g(XY) = g(YX)$ for all matrices $X, Y$ in $\mathbf{M}_n(C)$, there exists one and only one scalar $c \in C$ such that $g(X) = c \cdot \operatorname{Tr}(X)$ for every matrix $X \in \mathbf{M}_n(C)$.

Since the proposition is trivial for $n = 1$, attention may be confined to the case where $n \geqslant 2$. Taking $X = E_{ij},\ Y = E_{jk}$ with $i \neq k$, we obtain $g(E_{ik}) = 0$; then taking $X = E_{ij},\ Y = E_{ji}$ with $i \neq j$, we find $g(E_{ii}) = g(E_{jj})$; the proposition follows immediately since the $E_{ij}$ form a basis of $\mathbf{M}_n(\mathbf{C})$.