I am only beginning my study of group representations and characters. So far I have already encountered the regular group algebra $FG$. Although in an FG-module the multiplication is only defined for elements of the group $G$, it has unique linear extension, so in fact the regular module $FG$ acts on any $FG$-module, see e.g. Gordon-Liebeck, Representations and Characters of Groups p.57. And there are several situations where it is useful that we can use elements from $FG$.
At the moment I am starting to read about characters in the book I've mentioned above. Characters are functions from $G$ to $\mathbb C$. Of course, any such function has unique linear extension to a linear function $\mathbb C G\to\mathbb C$ defined on the regular $\mathbb C G$-module. So I was wondering, whether there are situations, where this extension comes handy.
Are group characters sometimes studied as linear maps on the regular $\mathbb C G$-module $\mathbb C G$? Are there some situations, where this approach can simplify things?