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Concept
Group algebra of a locally compact group
Summary
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 μ called a 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
The fact that is continuous is immediate from the dominated convergence theorem. Also
where the dot stands for the product in G. Cc(G) also has a natural involution defined by:
where Δ is the modular function on G. With this involution, it is a *-algebra.
Theorem. With the norm:
Cc(G) becomes 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
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 C[G].
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
is a non-degenerate bounded *-representation of the normed algebra Cc(G).
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