Group Algebra - The Group C*-algebra C*(G)

The Group C*-algebra C*(G)

Let C be the group ring of a discrete group G.

For a locally compact group G, the group C*-algebra C*(G) of G is defined to be the C*-enveloping algebra of L1(G), i.e. the completion of C_c(G) with respect to the largest C*-norm:

where π ranges over all non-degenerate *-representations of C_c(G) on Hilbert spaces. When G is discrete, it follows from the triangle inequality that, for any such π, π(f) ≤ ||f||₁. So the norm is well-defined.

It follows from the definition that C*(G) has the following universal property: any *-homomorphism from C to some B (the C*-algebra of bounded operators on some Hilbert space ) factors through the inclusion map C C*_max(G).