In functional analysis and related areas of mathematics, the **group algebra** is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.

If *G* is a locally compact Hausdorff group, *G* carries an essentially unique left-invariant countably additive Borel measure *μ* called a Haar measure. Using the Haar measure, one can define a convolution operation on the space *C _{c}*(

To define the convolution operation, let *f* and *g* be two functions in *C _{c}*(

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

where the dot stands for the product in *G*. *C _{c}*(

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

Theorem.With the norm:

C(_{c}G) becomes 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

Then {*f _{V}*}

Note that for discrete groups, *C _{c}*(

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

Theorem.LetGbe a locally compact group. IfUis a strongly continuous unitary representation ofGon a Hilbert spaceH, 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

Gand non-degenerate bounded *-representations ofC(_{c}G). This bijection respects unitary equivalence and strong containment. In particular, π_{U}is irreducible if and only ifUis irreducible.

Non-degeneracy of a representation π of *C _{c}*(

is dense in *H*_{π}.

It is a standard theorem of measure theory that the completion of *C _{c}*(

Theorem.L^{1}(G) is a Banach *-algebra with the convolution product and involution defined above and with theL^{1}norm.L^{1}(G) also has a bounded approximate identity.

Let **C**[*G*] 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 *L*^{1}(*G*), i.e. the completion of *C _{c}*(

where π ranges over all non-degenerate *-representations of *C _{c}*(

hence the norm is well-defined.

It follows from the definition that *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 *C _{r}**(

where

is the *L*^{2} norm. Since the completion of *C _{c}*(

Equivalently, *C _{r}**(

In general, *C _{r}**(

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 ℓ^{2}(*G*) for which *G* is an orthonormal basis. Since *G* operates on ℓ^{2}(*G*) by permuting the basis vectors, we can identify the complex group ring **C**[*G*] with a subalgebra of the algebra of bounded operators on ℓ^{2}(*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.