Group algebra

In mathematics, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i.e. a discrete group) a ring or algebra, such that the group multiplication induces the multiplication in the ring or algebra. As such, they are similar to the group ring associated to a discrete group.


Group algebra of a finite group

Given a finite group G, define the group algebra CG as the vector space over the complex numbers, with basis vectors <math>\{e_g\}<math> corresponding to the elements <math>g\in G<math>. The algebra structure on this vector space is defined as

<math>e_g \cdot e_h = e_{gh}<math>.

A representation of the algebra CG on a vector space V is the algebra homomorphism

<math>\mathbb{C} G\rightarrow \mbox{End} (V)<math>.

That is, a representation is a left CG-module. Any group representation <math>\rho:G\rightarrow \mbox{Aut}(V)<math> then extends linearly to an algebra representation <math>\overline{\rho}:\mathbb{C}G\rightarrow \mbox{End}(V)<math>. Thus, representations of the group correspond exactly to representations of the algebra, and so, in a certain sense, talking about the one is the same as talking about the other.

The center of the group algebra is the set of vectors which commute with the action of the group G on the vector space V:

<math>Z(\mathbb{C} G) := \left\{ z \in \mathbb{C} G \mid vzr = vrz \mbox{ for all } v \in V, z, r \in \mathbb{C} G\right\}<math>

Group algebras of topological groups: Cc(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 Cc(G) of complex-valued functions on G with compact support; Cc(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 Cc(G). For t in G, define

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

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

<math> \operatorname{Support}(f * g) \subseteq \operatorname{Support}(f) \cdot \operatorname{Support}(g) <math>

Cc(G) also has a natural involution defined by:

<math> f^*(s) = \overline{f(s^{-1})} \Delta(s^{-1}) <math>

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

Theorem. If Cc(G) is given the norm

<math> \|f\|_1 := \int_G |f(s)| d\mu(s), \quad <math> 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 fV be a non-negative continuous function supported in V such that

<math> \int_V f(g)\, d \mu(g) =1. \quad<math>

Then {fV}V is an approximate identity.

Note that for discrete groups, Cc(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

<math> \pi_U (f) = \int_G f(g) U(g)\, d \mu(g) \quad <math>

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

<math> U \mapsto \pi_U \quad <math>

is a bijection between the set of strongly continuous unitary representation of G and non-degenerate bounded *-representations of Cc(G). This bijection respects unitary equivalence and strong containment. In particular, πU is irreducible iff U is irreducible.

Non-degeneracy of a representation π of Cc(G). on a Hilbert space Hπ means that

<math> \{\pi(f) \xi: f \in \operatorname{C}_C(G), \xi \in H_\pi \} <math>

is dense in Hπ.

The convolution algebra L1(G)

It is a standard theorem of measure theory that the completion of Cc(G) in the L1(G) norm is isomorphic to the space L1(G) of functions which are integrable with respect to the Haar measure.

Theorem. L1(G) is a B*-algebra with the convolution product and involution defined above and with the L1 norm. L1(G) also has an approximate identity.

The group C*-algebra C*(G)

For a locally compact group G, the group C*-algebra of G is defined to be the C*-enveloping algebra of L1(G). It can also be defined as the completion of Cc(G) with respect to the norm

<math> \|f\|_{C^*} := \sup_\pi \|\pi(f)\| \quad <math>

where π ranges over all non-degenerate *-representations of Cc(G) on Hilbert spaces.

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

The reduced group C*-algebra focuses on the left regular representation of G rather than on all unitary representations of G. We thus consider the completion of Cc(G) with respect to the norm

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


<math> \|f\|_2 = \sqrt{\int_G |f|^2 d\mu} \quad <math>

is the L2 norm. Since the completion of Cc(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.

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.ja:群環


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