Given an arbitrary $\mathbb{C}$-algebra $A$, what should be required from a function $f:A\to A$ for it to be an anti-linear anti-involution?
As I understand “anti-linear” for $A$ as a vector space means that \begin{gather} \forall a_{1},a_{2}\in A: f(a_{1} + a_{2}) = f(a_{1}) + f(a_{2})\\ \forall a\in A\forall\lambda\in\mathbb{C}: f(\lambda a) = \overline{\lambda} f(a) \end{gather}
However I don’t still understand if it preserves product or not.
I know that for a Lie-algebra $\mathfrak{g}$ representation with highest weight anti-linear anti-involution also changes the order of elements in a bracket ($f([x,y]) = [f(y),f(x)]$), but I’m not sure how it acts on $U(\mathfrak{g})$, for example.