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. Theorem. With the norm:Theorem. Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, thenTheorem. 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. 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. In particular, these all use convolution to extend the group Hopf algebra from the set of functions with finite support to more useful classes of functions. 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 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 f * g 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. 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.