a sum of hermitian squares.

\[rem2.4\] The two statements in the PS Conjecture are indeed equivalent: the necessary direction follows from the fact that traces of hermitian squares are totally ${\sigma}$-positive, as observed in Example \[ex2.2\].

For the sufficient direction, assume that the trace of a hermitian square is always a sum of hermitian squares. Let $a\in \operatorname{Sym}(A,{\sigma})$ be totally ${\sigma}$-positive. Then $a$ can be expressed in terms of the entries in a diagonalization of the form $\operatorname{Trd}({\sigma}(x)x)$ as in Theorem \[thm:NCh17\](ii). Let $\beta$ be such an entry. Thus, $\beta=\operatorname{Trd}({\sigma}(y)y)$ for some $y\in A$. By the assumption there are $x_1,\ldots, x_\ell\in A$ such that $\beta=\sum_i {\sigma}(x_i)x_i$. Since $\beta\in F$, the expression in Theorem \[thm:NCh17\](ii) can now be rewritten as a sum of hermitian squares.

As mentioned a few lines earlier, Procesi and Schacher provide supporting evidence for their conjecture for the case $\deg A=2$. Another case where the PS Conjecture is true has been well-known since the 1970s:

\[ex:psd\] Let $A$ be the full matrix ring $M_n(F)$ over a formally real field $F$ endowed with the transpose involution ${\sigma}=t$. Since $\operatorname{Trd}=\operatorname{tr}$, every ordering of $F$ is a $\sigma$-ordering. We claim that $a\in\operatorname{Sym}(A,\sigma)$ is totally ${\sigma}$-positive if and only if $a$ 