Does $|x^*|=|x|$ in a star ring with an absolute value?

Question

Let $R$ be a star ring with an absolute value. Is it true that $|x^*|=|x|$ for all $x\in R$?

Here a star ring is a ring with a function $*:R\to R$ called conjugation such that

• $(x+y)^*=x^*+y^*$
• $(xy)^*=y^*x^*$
• $x^{**}=x,$

and an absolute value is a function $|\cdot|:R\to\Bbb R$ such that

• $|x|=0\iff x=0$
• $|x-y|\le|x|+|y|$
• $|xy|=|x||y|.$

Obviously it is true for the trivial conjugation $x^*=x$, and it is also true for $\Bbb C$ and matrix rings over $\Bbb R$ and $\Bbb C$ with transposition and any of the various common matrix norms, so I wonder if it is true in general.

Answer 1

This is not necessarily the case. For example, $\mathbb{Q}(\sqrt{17})$ forms a star ring with absolute value under the conjugation $(a+b\sqrt{17})^*=a-b\sqrt{17}$ and the standard (Euclidean) absolute value, but certainly $|1+\sqrt{17}|\neq |1-\sqrt{17}|$.