Given a finite group G, its complexified ring of finite-dimensional complex representations is isomorphic to its algebra of class functions, by the trace map $\mathrm{Tr}_\rho: g \mapsto \mathrm{Tr}(\rho(g))$. These class functions can in turn can be shown to correspond to the elements in the center of the group algebra, by sending a class function $c:G \to \mathbb{C}$ to the element $\sum_{g \in G} c(g) \cdot g$ in the group algebra.

This map from class functions to the group algebra isn't a ring homomorphism. In particular, it doesn't preserve the unit. However, I'm pretty sure that the representation ring and the center of the group algebra are nevertheless isomorphic as algebras. Am I right? Even better, is there a canonical such homomorphism with some group-theoretical importance?