Group algebra

This page discusses topological algebras associated to topological groups; for the purely algebraic case of discrete groups see group ring.

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)

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 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

$[f * g](t) = \int_G f(s) g(s^{-1} t)\, d \mu(s). \quad$

The fact that f * g is continuous is immediate from the dominated convergence theorem. Also

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

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. If C_c(G) is given the norm

$\|f\|_1 := \int_G |f(s)| d\mu(s), \quad$ 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

$\int_V f_{V}(g)\, d \mu(g) =1. \quad$

Then {f_V}_V is an approximate identity. A group algebra can only have an identity, as opposed to just 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 CG.

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

$\pi_U (f) = \int_G f(g) U(g)\, d \mu(g) \quad$

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

$U \mapsto \pi_U \quad$

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

$\{\pi(f) \xi: f \in \operatorname{C}_c(G), \xi \in H_\pi \}$

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 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:

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

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, π(f) ≤ ||f||₁. So 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($\mathcal{H}$) (the C*-algebra of bounded operators on some Hilbert space $\mathcal{H}$) factors through the inclusion map 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

$\|f\|_{C^*_r} := \sup \{ \|f*g\|_2: \|g\|_2 = 1\}, \quad$

where

$\|f\|_2 = \sqrt{\int_G |f|^2 d\mu} \quad$

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 "convolution by f" acting on L²(G) 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 CG 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.

See also

• Graph algebra
• Incidence algebra
• Path algebra
• Group ring

References

• J, Dixmier, C* algebras, ISBN 0-7204-0762-1
• A. A. Kirillov, Elements of the theory of representations, ISBN 0-387-07476-7
• A.I. Shtern (2001), "Group algebra of a locally compact group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/G/g045230.htm 

This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

• L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30