# Group algebra of a locally compact group

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
    $[f*g](t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).$
The fact that $f*g$ is continuous is immediate from the dominated convergence theorem. Also
    $\operatorname {Support} (f*g)\subseteq \operatorname {Support} (f)\cdot \operatorname {Support} (g)$
where the dot stands for the product in G. C$_c$(G) also has a natural involution defined by:
    $f^{*}(s)={\overline {f(s^{-1})}}\,\Delta (s^{-1})$
where Δ is the modular function on G. With this involution, it is a *-algebra.
   Theorem. With the norm:
       $\|f\|_{1}:=\int _{G}|f(s)|\,d\mu (s),$
   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
    $\int _{V}f_{V}(g)\,d\mu (g)=1.$
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
       $\pi _{U}(f)=\int _{G}f(g)U(g)\,d\mu (g)$
   is a non-degenerate bounded *-representation of the normed algebra C$_c$(G). The map
       $U\mapsto \pi _{U}$
   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
    $\left\{\pi (f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\}$
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 the L$^1$ 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:
    $\|f\|_{C^{*}}:=\sup _{\pi }\|\pi (f)\|,$
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:
    $\|\pi (f)\|\leq \|f\|_{1},$
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:
    $\mathbf {C} [G]\hookrightarrow C_{\max }^{*}(G).$

## 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
    $\|f\|_{C_{r}^{*}}:=\sup \left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},$
where
    $\|f\|_{2}={\sqrt {\int _{G}|f|^{2}\,d\mu }}$
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 acting on 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.