Tuesday, 24 September 2013

Group Algebras of Topological Groups

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: 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)
were the dot stands for the product in 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 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, 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.

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