# Group algebra

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In 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.

## Group algebras of topological groups: *C*_{c}(*G*)

_{c}

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 a 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

were the dot stands for the product in *G*. *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.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

*V*such that

Then {*f _{V}*}

_{V}is an approximate identity. A group algebra has an identity, as opposed to just an 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

**C**[

*G*].

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, thenis a non-degenerate bounded *-representation of the normed algebra

C(_{c}G). The mapis 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}*(

*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 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 only on a set of Haar measure zero.

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.

## The group C*-algebra *C**(*G*)

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}*(

*G*) with respect to the largest C*-norm:

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

*G*) on Hilbert spaces. When

*G*is discrete, it follows from the triangle inequality that, for any such π, one has:

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}*(*G*)

_{r}*

The reduced group C*-algebra *C _{r}**(

*G*) is 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*norm is the norm of the bounded operator acting on

_{r}**L*

^{2}(

*G*) by convolution with

*f*and thus a C*-norm.

Equivalently, *C _{r}**(

*G*) is the C*-algebra generated by the image of the left regular representation on ℓ

^{2}(

*G*).

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

*G*) is a quotient of

*C**(

*G*). 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.

## See also

## References

- J, Dixmier,
*C* algebras*, ISBN 0-7204-0762-1 - A. A. Kirillov,
*Elements of the theory of representations*, ISBN 0-387-07476-7 - L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30
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*This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*