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.

The algebra C_c(G) of continuous functions with compact support

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(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

${\displaystyle [f*g](t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).}$

The fact that ${\displaystyle f*g}$ is continuous is immediate from the dominated convergence theorem. Also

${\displaystyle \operatorname {Support} (f*g)\subseteq \operatorname {Support} (f)\cdot \operatorname {Support} (g)}$

where the dot stands for the product in G. C_c(G) also has a natural involution defined by:

${\displaystyle f^{*}(s)={\overline {f(s^{-1})))\,\Delta (s^{-1})}$

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

Theorem. With the norm:

${\displaystyle \|f\|_{1}:=\int _{G}|f(s)|\,d\mu (s),}$

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

${\displaystyle \int _{V}f_{V}(g)\,d\mu (g)=1.}$

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. Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, then

${\displaystyle \pi _{U}(f)=\int _{G}f(g)U(g)\,d\mu (g)}$

is a non-degenerate bounded *-representation of the normed algebra C_c(G). The map

${\displaystyle U\mapsto \pi _{U))$

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

${\displaystyle \left\{\pi (f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\))$

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 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. L¹(G) is a Banach *-algebra with the convolution product and involution defined above and with the L¹ norm. L¹(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 L¹(G), i.e. the completion of C_c(G) with respect to the largest C*-norm:

${\displaystyle \|f\|_{C^{*)):=\sup _{\pi }\|\pi (f)\|,}$

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:

${\displaystyle \|\pi (f)\|\leq \|f\|_{1},}$

hence the norm is well-defined.

It follows from the definition that, when G is a discrete group, 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:

${\displaystyle \mathbf {C} [G]\hookrightarrow C_{\max }^{*}(G).}$

The reduced group C*-algebra C_r*(G)

The reduced group C*-algebra C_r*(G) is the completion of C_c(G) with respect to the norm

${\displaystyle \|f\|_{C_{r}^{*)):=\sup \left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},}$

where

${\displaystyle \|f\|_{2}={\sqrt {\int _{G}|f|^{2}\,d\mu ))}$

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 acting on L²(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 ℓ²(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 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 C[G] 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 II₁ factor if and only if G is countable, amenable, and has the infinite conjugacy class property.