Group Algebra - Group Algebras of Topological Groups: C_c(G)

Group Algebras of Topological Groups: C_c(G)

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called Haar measure. Using the Haar measure, one can define a convolution operation on the space C_c(G) of complex-valued continuous functions on G with compact support; C_c(G) can then be given any of various norms and the completion will be a group algebra.

To define the convolution operation, let f and g be two functions in C_c(G). For t in G, define

The fact that f * g is continuous is immediate from the dominated convergence theorem. Also

C_c(G) also has a natural involution defined by:

where Δ is the modular function on G. With this involution, it is a *-algebra.

Theorem. If C_c(G) is given the norm

it becomes is an involutive normed algebra with an approximate identity.

The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed if V is a compact neighborhood of the identity, let f_V be a non-negative continuous function supported in V such that

Then {f_V}_V is an approximate identity. A group algebra can only have an identity, as opposed to just approximate identity, if and only if the topology on the group is the discrete topology.

Note that for discrete groups, C_c(G) is the same thing as the complex group ring CG.

The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following

Theorem. Let G be a locally compact group. 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 C_c(G). The map

is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded *-representations of C_c(G). This bijection respects unitary equivalence and strong containment. In particular, π_U is irreducible if and only if U is irreducible.

Non-degeneracy of a representation π of C_c(G) on a Hilbert space H_π means that

is dense in H_π.