Group algebra

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.

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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 continuous 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

 [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)

Cc(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 Cc(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 fV be a non-negative continuous function supported in V such that

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

Then {fV}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, 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

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

is a non-degenerate bounded *-representation of the normed algebra Cc(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 Cc(G). This bijection respects unitary equivalence and strong containment. In particular, πU is irreducible if and only if U is irreducible.

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

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

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 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 Cc(G) with respect to the largest C*-norm:

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

where π ranges over all non-degenerate *-representations of Cc(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 Cc(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 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.

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

References

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

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