### Group Algebras of Topological Groups

In mathematics, the

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Group algebras of topological groups:

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups

To define the convolution operation, let

Note that for discrete groups,

The importance of the group algebra is that it captures the unitary representation theory of

Non-degeneracy of a representation π of

**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*is continuous is immediate from the dominated convergence theorem. Also*C*_{c}(*G*) also has a natural involution defined by:*G*. With this involution, it is a *-algebra.**Theorem**. If*C*_{c}(*G*) is given the norm- it becomes is an involutive normed algebra with an approximate identity.

*V*is a compact neighborhood of the identity, let*f*_{V}be a non-negative continuous function supported in*V*such that*f*_{V}}_{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,

*C*_{c}(*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*C*_{c}(*G*). The map*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*H*_{π}.**To Join Ajit Mishra's Online Classroom***CLICK HERE*
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