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Group algebra of a locally compact group
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Group algebra of a locally compact group
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.
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Group algebra of a locally compact group
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.