There is a direct way to define group algebras for different classes of groups (locally compact groups, Lie groups, algebraic groups, etc.). 

For an (affine) algebraic group $G$ one should take the algebra ${\mathcal R}(G)$ of polynomials (i.e. regular functions) on $G$ and then consider the dual algebra ${\mathcal R}^\star(G)$ of linear functionals on ${\mathcal R}(G)$ with the natural topology (in this case the topology of poitwise convergence on ${\mathcal R}(G)$). Then ${\mathcal R}^\star(G)$ becomes a (topological) group algebra for $G$: each  "polynomial representation" of $G$ corresponds bijectively to a "polynomial representation" of ${\mathcal R}^\star(G)$. At the same time both ${\mathcal R}(G)$ and ${\mathcal R}^\star(G)$ are Hopf algebras in the monoidal category of stereotype spaces, see details in: S.S.Akbarov, Pontryagin duality in the theory of topological vector spaces and in topological algebra, *Journal of Mathematical Sciences*, 113(2): 179-349, 2003 (Theorem 10.12).