I have been studying $C^*$-algebras and operator theory in general and I have some questions.
A common example of a $C^*$-algebra is $M_n(\mathbb C)$ the $n\times n$ complex matrices. Since $M_n(\mathbb R) \subseteq M_n(\mathbb C)$, the results regarding the latter can also be applied to real matrices, however the results may not always concern real matrices.
For example, in a $C^*$-algebra $A$, an element $a \in A$ is said to be positive if it is self-adjoint and its spectrum is non-negative. It is also a known result that this is equivalent to saying there is some $q \in A$ such that $a = q^*q$. Thus one could take $A= M_n(\mathbb C)$, a positive semidefinite real matrix $a \in A$ and apply the result. However, there are no guarantees that $q$ must also be a real matrix.
Even so, it can be proven that it holds, i.e., that a matrix $a$ is positive semidefinite if and only if there is a real matrix $q$ such that $a = q^* q = q^Tq$.
My question is then, to what extent do general results regarding $C^*$-algebras hold when considering real matrices.
Perhaps each case is a case, or maybe there is a general rule of thumb one can use? Or even a whole area that analyses these cases. I've seen some stuff that talked about ``real $C^*$-algebras'', however, I didn't seem to find something regarding its groundwork. Either way, any response is appreciated.