When an algebra isomorphism preserves positive involution

Question

Let $A$ be a $K$-algebra 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 there is a $K$-algebra isomorphism $$\varphi: A \to B.$$

If there is a unique (up to isomorphism) positive involution in $A$ and $B$, can we say the involution must be preserved by the isomorphism $\varphi$: $$\varphi: (A, *_A) \to (B, *_B)?$$

Motivation:

My question is motivated from representation theory. When $W$ is a real irreducible representation (irrep) of a finite group $G$, then $D:=End_G(W)$ is a division algebra by Schur's lemma. Additionally, Frobenius theorem on real associative division algebras implies that $D$ is isomorphic to either the reals $\mathbb{R}$, complexes $\mathbb{C}$, or quaternions $\mathbb{H}$. Let's call this isomorphism $\varphi$.

Suppose $W$ is endowed with an $G$-invariant inner product (i.e. $\langle g\cdot u, g\cdot v \rangle=\langle u , v \rangle$ for all $u,v \in W$ and $g \in G$). So, $W$ is an orthogonal irrep, and $D$ has the involution that acts as the adjoint corresponding to the $G$-invariant inner product. We know that $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$ all have involution. Furthermore, they are normed division algebras. The involution is the trivial map in $\mathbb{R}$, complex conjugate in $\mathbb{C}$, and the standard involution in $\mathbb{H}$ $\left((a+ib+jc+dk)^*=a-ib-jc-dk\right).$ We even do not need to define the involution in $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$ because the involution must be positive when we want to correspond involution to adjoint and there is a unique positive involution up to isomorphism in real central division algebras. My original question is how to prove that $\varphi$ preserves involution. Is this even true?

Answer 1

The answer to your general question is NO.

Assume for simplicity that $A=B$, and that $A$ is a central simple $K$-algebra. Then $\varphi$ is an inner automorphism by Skolem Noether 's Theorem.

Write $\varphi=Int(a)$ (conjugation by $a$) for some $a\in A^\times$. Then $\varphi$ preserves the involution if and only if $(axa^{-1})^*=a^{-*}x^*a^*=ax^*a^{-1}$ for all $x\in A$. This means that $a^*a$ is central, that is $a^*a\in K^\times$.

This does not hold for all $a\in A$, in general, even if $*$ is positive.

You can find concrete examples by setting $K=\mathbb{R},$ $A=B=M_n(\mathbb{R})$, $*=$ transposition of matrices, and $a$ any invertible matrix such that $a^ta$ is not a scalar multiple of the identity matrix.