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.
The algebra Cc(G) of continuous functions with compact support 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
[ f ∗ g ] ( t ) = ∫ G f ( s ) g ( s − 1 t ) d μ ( s ) . {\displaystyle [f*g](t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).}
The fact that f ∗ g {\displaystyle f*g} is continuous is immediate from the dominated convergence theorem. Also
Support ( f ∗ g ) ⊆ Support ( f ) ⋅ Support ( g ) {\displaystyle \operatorname {Support} (f*g)\subseteq \operatorname {Support} (f)\cdot \operatorname {Support} (g)}
where the dot stands for the product in G. Cc(G) also has a natural involution defined by:
f ∗ ( s ) = f ( s − 1 ) ¯ Δ ( s − 1 ) {\displaystyle 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. With the norm:
‖ f ‖ 1 := ∫ G | f ( s ) | d μ ( s ) , {\displaystyle \|f\|_{1}:=\int _{G}|f(s)|\,d\mu (s),}
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
∫ V f V ( g ) d μ ( g ) = 1. {\displaystyle \int _{V}f_{V}(g)\,d\mu (g)=1.}
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
π U ( f ) = ∫ G f ( g ) U ( g ) d μ ( g ) {\displaystyle \pi _{U}(f)=\int _{G}f(g)U(g)\,d\mu (g)}
is a non-degenerate bounded *-representation of the normed algebra Cc(G). The map
U ↦ π U {\displaystyle U\mapsto \pi _{U}}
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
{ π ( f ) ξ : f ∈ C c ( G ) , ξ ∈ H π } {\displaystyle \left\{\pi (f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\}}
is dense in Hπ.
The convolution algebra L1(G) It is a standard theorem of measure theory that the completion of Cc(G) in the L1(G) norm is isomorphic to the space L1(G) of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero.
Theorem. 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.
… excerpt ends here. Continue reading the full article.
