Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.

Suppose that $A$ is a finite-dimensional vector space over a field $k$ with $\operatorname{char}(k) = 0$, equipped with:

• a commutative associative $k$-linear multiplication $\circ\\,$;
• a positive-definite inner product $\langle \cdot , \cdot \rangle \\,$.

For each $a \in A$, let $L_a \colon A \to A \colon x \mapsto a \circ x$. This is a linear operator on $A$; consider the adjoint operator w.r.t. the given inner product:

$$\langle L_a x, y \rangle = \langle x, L_a^* y \rangle$$ for all $x,y \in A$. Now write $y \star a := L_a^* y$.

Popescu's question is the following:

Is there a name for this operation $\star$? Has it been studied in the literature? Is there anything known about algebraic properties or identities involving $\star$ (possibly also involving the other data $\circ$ and $\langle \cdot , \cdot \rangle$)?