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 an ordered 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 $\langle \cdot , \cdot \rangle$)?
> 
> 
> 



adjoint of multiplication operator in a commutative algebra
In case $k=\mathbb{C}$, what you're describing is a finite-dimensional *H\*-algebra*. More generally, these are Banach algebras, whose carrier space is a Hilbert space, satisfying the adjoint property you mention. 


It is natural to make a nondegeneracy assumption: $A$ is called *proper* when $\forall a \in A\\,.\\, a \circ A = 0 \Rightarrow a = 0$. This turns out to be equivalent to the adjoint $L\_a^\ast$ being unique, or, in other words, $\ast$ being an involution. Every H\*-algebra is a direct sum of a proper one and an algebra in which $a \circ b=0$ for all $a$ and $b$.


There is a neat structure theorem by Warren Ambrose (see <http://www.jstor.org/stable/1990182>), showing that proper H\*-algebras are always direct sums of full matrix algebras. In particular, commutative H\*-algebras are direct sums of 1-dimensional algebras, and hence correspond precisely to orthogonal basis of their carrier space! 


I don't know about other fields $k$, but Ambrose's proof basically comes down to carefully analyzing idempotents, which is feasible to repeat for other fields $k$.


