We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such that $J^2=-I$ are matrix representations of $\mathbb{C}$. This means that any matrix of the form $$ J=\left[ \begin {array}{cccc} a&b\\ c&-a \end{array} \right] \qquad bc \le-1 \qquad a=\sqrt{-1-bc} $$ is a possible representation of the imaginary unity $i$ and a complex number $z=x+iy$ is represented by the matrix $M(z)=Z=xI+yJ$ I think that this fact can be see as a consequence of the Artin–Wedderburn theorem about the matrix representation of rings ( see my answer to: What does it mean to represent a number in term of a $2\times2$ matrix?), that generalize such kind of representation to all semisimple rings.

Now, consider $\mathbb{C}$ as a $^*$Algebra (with complex conjugation as the involution). If we search a $^*$Algebra isomorphism such that $M(\bar z)=M^T(z)$ (where the involution for the matrix ring is the transpose), we see that there are only two possible representations, with :

$$ J=\left[ \begin {array}{cccc} 0&1\\ -1&0 \end{array} \right] \qquad \mbox{or}\qquad J=\left[ \begin {array}{cccc} 0&-1\\ 1&0 \end{array} \right] $$ So we have a great restriction of the possible representations.

My question is if this restriction is a consequence of some general representation theorem for *Algebras in the same sense as the matrix representation of the field $\mathbb{C}$ is a consequence of the A-W theorem.


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