of $A$ is clearly a subset of $\operatorname{Sym}(A,{\sigma})$. It is also clear that the hermitian squares of $K$ are all in $F$.

\[ex2.2\] Sums of hermitian squares and sums of traces of hermitian squares are examples of totally ${\sigma}$-positive elements, as easy verifications will show.

One of the main results in [@PS] explains that these are essentially the only examples. It can be considered as a noncommutative analogue of Artin’s solution to Hilbert’s 17th problem:

[[@PS Theorem 5.4]]{} \[thm:NCh17\] Let $A$ be a central simple algebra with involution ${\sigma}$, centre $K$ and fixed field $F$. Let $\alpha_1,\ldots,\alpha_m\in F$ be elements appearing in a diagonalisation of the quadratic form $\operatorname{Trd}(\sigma(x)x)$. Then for $a\in\operatorname{Sym}(A,\sigma)$ the following statements are equivalent:

1.  $a$ is totally ${\sigma}$-positive;

2.  there exist $x_{i,{\varepsilon}}\in A$ with $$a=\sum_{{\varepsilon}\in \{0,1\}^m} \alpha^{\varepsilon}\sum_{i} \sigma(x_{i,{\varepsilon}}) 
    x_{i,{\varepsilon}}.$$ $($As usual, $\alpha^{\varepsilon}$ denotes $\alpha_1^{{\varepsilon}_1}\cdots \alpha_m^{{\varepsilon}_m}$.$)$

In the case $n=\deg A=2$, the weights $\alpha_j$ are superfluous (we will come back to this later). Procesi and Schacher [@PS p. 404] conjecture that this is also the case for $n>2$:

\[open:PS\] In a central simple algebra $A$ with involution ${\sigma}$, every totally ${\sigma}$-positive element is a sum of hermitian squares. Equivalently: the trace of a hermitian square is always 