The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following Theorem.
If U is a strongly continuous unitary representation of G on a Hilbert space H, then is a non-degenerate bounded *-representation of the normed algebra Cc(G).
is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded *-representations of Cc(G).
It follows from the definition that, when G is a discrete group, C*(G) has the following universal property: any *-homomorphism from C[G] to some B(H) (the C*-algebra of bounded operators on some Hilbert space H) factors through the inclusion map: The reduced group C*-algebra Cr*(G) is the completion of Cc(G) with respect to the norm where is the L2 norm.
Equivalently, Cr*(G) is the C*-algebra generated by the image of the left regular representation on ℓ2(G).
The weak closure of this subalgebra, NG, is a von Neumann algebra.
NG is isomorphic to the hyperfinite type II1 factor if and only if G is countable, amenable, and has the infinite conjugacy class property.