# Group algebra

In mathematics, 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 or algebra, 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.

## Contents

## Group algebra of a finite group

Given a finite group *G*, define the group algebra *CG* as the vector space over the complex numbers, with basis vectors corresponding to the elements . The algebra structure on this vector space is defined as

- .

A representation of the algebra *CG* on a vector space *V* is the algebra homomorphism

- .

That is, a representation is a left *CG*-module. Any group representation then extends linearly to an algebra representation . Thus, representations of the group correspond exactly to representations of the algebra, and so, in a certain sense, talking about the one is the same as talking about the other.

The *center* of the group algebra is the set of vectors which commute with the action of the group *G* on the vector space *V*:

## 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 **C***G*.

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 *L*^{1}(*G*)

It is a standard theorem of measure theory that the completion of *C*_{c}(*G*) in the *L*^{1}(*G*) norm is isomorphic to the space *L*^{1}(*G*) of functions which are integrable with respect to the Haar measure.

**Theorem**. *L*^{1}(*G*) is a B*-algebra with the convolution product and involution defined above and with the *L*^{1} norm. *L*^{1}(*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 *L*^{1}(*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.

Let be the group ring of a discrete group It has the following two completions to a -algebra:

### Reduced group -algebra

The reduced group -algebra, , is obtained by completing in the operator norm for its regular representation on

### Maximal group -algebra

The maximal group -algebra, or just , is defined by the following universal property: any *-homomorphism from to some (the -algebra of bounded operators on some Hilbert space ) factors through the inclusion map

If is amenable then

## 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 L^{2} norm. Since the completion of *C*_{c}(*G*) with regard to the L^{2} norm is a Hilbert space, the C^{*}_{r} norm is the norm of the bounded operator convolution by *f* acting on *L*^{2}(*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 *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.

*This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the GFDL.*