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 ü called a Haar measure. Using the Haar measure, one can define a convolution operation on the space C<sub>c</sub>(G) of complex-valued continuous functions on G with compact support; C<sub>c</sub>(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<sub>c</sub>(G). For t in G, define
The fact that is continuous is immediate from the dominated convergence theorem. Also
where the dot stands for the product in G. C<sub>c</sub>(G) also has a natural involution defined by:
where ÃÂ is the modular function on G. With this involution, it is a *-algebra.
<blockquote>Theorem. With the norm:
C<sub>c</sub>(G) becomes an involutive normed algebra with an approximate identity.</blockquote>
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<sub>V</sub> be a non-negative continuous function supported in V such that
Then {f<sub>V</sub>}<sub>V</sub> 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<sub>c</sub>(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
<blockquote>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<sub>c</sub>(G). The map
is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded *-representations of C<sub>c</sub>(G). This bijection respects unitary equivalence and strong containment. In particular, <sub>U</sub> is irreducible if and only if U is irreducible.</blockquote>
Non-degeneracy of a representation of C<sub>c</sub>(G) on a Hilbert space H<sub></sub> means that
is dense in H<sub></sub>.
It is a standard theorem of measure theory that the completion of C<sub>c</sub>(G) in the L<sup>1</sup>(G) norm is isomorphic to the space L<sup>1</sup>(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.
<blockquote>Theorem. L<sup>1</sup>(G) is a Banach *-algebra with the convolution product and involution defined above and with the L<sup>1</sup> norm. L<sup>1</sup>(G) also has a bounded approximate identity.</blockquote>
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<sup>1</sup>(G), i.e. the completion of C<sub>c</sub>(G) with respect to the largest C*-norm:
where ranges over all non-degenerate *-representations of C<sub>c</sub>(G) on Hilbert spaces. When G is discrete, it follows from the triangle inequality that, for any such , one has:
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:
The reduced group C*-algebra C<sub>r</sub>*(G) is the completion of C<sub>c</sub>(G) with respect to the norm
where
is the L<sup>2</sup> norm. Since the completion of C<sub>c</sub>(G) with regard to the L<sup>2</sup> norm is a Hilbert space, the C<sub>r</sub>* norm is the norm of the bounded operator acting on L<sup>2</sup>(G) by convolution with f and thus a C*-norm.
Equivalently, C<sub>r</sub>*(G) is the C*-algebra generated by the image of the left regular representation on âÂÂ<sup>2</sup>(G).
In general, C<sub>r</sub>*(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.
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 âÂÂ<sup>2</sup>(G) for which G is an orthonormal basis. Since G operates on âÂÂ<sup>2</sup>(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 âÂÂ<sup>2</sup>(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<sub>1</sub> factor if and only if G is countable, amenable, and has the infinite conjugacy class property.