Suppose $R$ is a ring with involution $*$ and $x,y\in R$. Does the quantity $xy-y^{*}x^{*}$ have a standard name? Has this product undergone systematic study in the ring-theory literature, and if so, where? (It may be, for fundamental reasons, that this product is not as interesting as the standard Lie product. If so, that is fine.)
2 Answers
For an element $x$ of a ring $R$ with involution $\ast$, the element $x-x^\ast$ is called the skew-trace of $x$. (See e.g. Chapter 2 in the Herstein's book "Rings with involution"). So $xy-y^\ast x^\ast=xy-(xy)^\ast$ is just the skew-trace of $xy$.
A simple remark:
If $(xy)^*= y^*x^*$ then this is just $xy-(xy)^*$ which is "twice the imaginary part of the product". When $R=\mathbb C$ then this is up to a factor the symplectic form on $\mathbb R^2$ and the determinant of $$ \begin{pmatrix} \Re x & \Re y \\ \Im x & \Im y \end{pmatrix} $$
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$\begingroup$ Thanks, Peter! I'm interested in this "imaginary part" interpretation...as these quantities have arisen in connection with studying the complex group algebra as a *-algebra. I was not aware of this symplectic form. $\endgroup$ Mar 18, 2016 at 15:00