I have asked this question: When an algebra isomorphism preserves positive involution, but now I want to modify it.
Let $A$ and $B$ be $K$-algebras where $K$ is a field with a unique ordering. We say a $K$-linear involution $*$ is positive if the map $A \to K$ via $a \mapsto tr(a^*a)$ is positive definite with respect to the ordering. Here, $tr(a)$ is the trace of the left multiplication map $a:x \mapsto ax$. Suppose
$$A \cong B $$ as $K$-algebras.
If there is a unique (up to isomorphism) positive involution in $A$ and in $B$, can we say that $$(A, *_A) \cong (B, *_B)?$$
