Group Algebra - The Convolution Algebra L1(G)

The Convolution Algebra L1(G)

It is a standard theorem of measure theory that the completion of C_c(G) in the L1(G) norm is isomorphic to the space L1(G) of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ on a set of Haar measure zero.

Theorem. L1(G) is a Banach *-algebra with the convolution product and involution defined above and with the L1 norm. L1(G) also has a bounded approximate identity.