under the canonical map ${\mathbb{K}}{\langle{\overline}X,{\overline}X^*\rangle}\to\operatorname{GM}_n({\mathbb{K}},\operatorname{J})$. Every $*$-homomorphism $\operatorname{GM}_n({\mathbb{K}},\operatorname{J})\to M_n(L)$ for a $*$-field extension $L$ of ${\mathbb{K}}$, where $M_n(L)$ is given a type $\operatorname{J}$ involution, yields a $*$-homomorphism ${\mathbb{K}}{\langle{\overline}X,{\overline}X^*\rangle}\to M_n(L)$, so is essentially given by a point $s\in M_n(L)^{\mathbb{N}}$ describing the images of the $X_i$ under this induces map.

By construction, the image $\beta_1\otimes 1$ of $\beta_1$ under the embedding of algebras with involution $\operatorname{GM}_n({\mathbb{K}},\operatorname{J})\to A$ is not $\sigma$-positive. Let $s$ denote the corresponding evaluation point. By Example \[ex:psd\], this means that $
\hat \beta_1(s,{\overline}{s}^t)=\beta_1\otimes 1$ is not positive semidefinite. Consider the following elementary statement: $$\label{eq:1st}
\begin{split}
\exists \,n\times n \text{ matrices }x=(x_1,\ldots, x_N) : \; &
\hat\alpha_i(x,{\overline}{x}^t) \text{ is positive semidefinite } \land \\
& \hat\beta_1(x,{\overline}{x}^t) \text{ is not positive semidefinite}.
\end{split}$$ ($N$ is the maximal number of variables appearing in one of the $\hat\alpha_i, \hat\beta_1$.)

Obviously such $n\times n$ matrices $x_i$ can be found over ${\overline}Z^{ {\rm rc} }$ or ${\overline}Z$; just take $x_i=s_i$. By Tarski’s transfer principle, the above elementary statement can be satisfied in ${\mathbb{K}}$. This yields a $*$-homomorphism ${\mathbb{K}}{\langle{\overline}X,{\overline}X^*\rangle}\to M_n({\mathbb{K}})$ endowed with the (positive) involution ${{\overline}{\phantom{x}}}t$ and in turn (by universality) a $*$-homomorphism $\operatorname{GM}_n({\mathbb{K}},\operatorname{J})\to (M_n({\mathbb{K}}),{{\overline}{\phantom{x}}}t)$. By the construction, the image of $a$ under this mapping will 