I am reading these two wiki articles: https://en.wikipedia.org/wiki/Group_algebra and https://en.wikipedia.org/wiki/Pontryagin_duality
From my understanding, the group algebra of a topological group G is the collection of all continuous functions $f : G \to \Bbb{C}$ with compact support, denoted by $C_c(G)$. When $G$ is discrete, compact support implies finite support, so we can identify $f : G \to \Bbb{C}$ with the sum $\sum_{g \in G} f(g) g$, which shows that the group ring and group algebra are isomorphic for discrete groups. Does this sound right?
However, I am confused by a passage in the second link given above:
This algebra [$L^1(G)$] is referred to as the Group Algebra of $G$
What is in fact the group algebra?