In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i.e. a discrete group) a ring (mathematics) or algebra over a field, such that the group multiplication induces the multiplication in the ring or algebra. As such, they are similar to the group ring associated to a discrete group.

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 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 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.

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 representation 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 iff 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.

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 functions which are integrable with respect to the Haar measure.

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

The group C*-algebra C*(G)

For a locally compact group G, the group C*-algebra of G is defined to be the C*-enveloping algebra of L¹(G). It can also be defined as the completion of C_c(G) with respect to the norm

where π ranges over all non-degenerate *-representations of C_c(G) on Hilbert spaces.

The reduced group C*-algebra C^*_r(G)

The reduced group C*-algebra focuses on the left regular representation of G rather than on all unitary representations of G. We thus consider the completion of C_c(G) with respect to the norm

where

is the L² norm. Since the completion of C_c(G) with regard to the L² norm is a Hilbert space, the C^*_r norm is the norm of the bounded operator convolution by f acting on L²(G) and thus a C* norm.

The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if G is amenable.

von Neumann algebras associated to groups

The group von Neumann algebra W*(G) of G is the enveloping von Neumann algebra of C*(G).

For a discrete group G, we can consider the Hilbert space l²(G) for which G is an orthonormal basis. Since G operates on l²(G) by permuting the basis vectors, we can identify the complex group ring CG with a subalgebra of the algebra of bounded operators on l²(G). The weak closure of this subalgebra, NG, is a von Neumann algebra.

The center of NG can be described in terms of those elements of G whose conjugacy class is finite. In particular, if the identity element of G is the only group element with that property (that is, G has the infinite conjugacy class property), the center of NG consists only of complex multiples of the identity.

NG is isomorphic to the hyperfinite type II-1 factor if and only if G is countable, amenable, and has the infinite conjugacy class property.