A Biased Motivation: Idempotents in Group Algebras

Abstract

Let us start with a countable group Γ. We linearize Γ by associating to it the complex group algebra CΓ, where CΓ is the C-vector space with basis Γ. It can also be viewed as the space of functions f: Γ → C with finite support. The product in CΓ is induced by the multiplication in Γ. Namely, for $f = \sum\nolimits_{8 \in \Gamma } {{f_s}s}$ and $g = \sum\nolimits_{t \in \Gamma } {{g_t}t}$ elements in CΓ, then

$$ f * g = \sum\limits_{s,t \in \Gamma } {{f_s}{g_t}st} $$

which is the usual convolution of f and g, and thus

$$ f * g\left( t \right) = \sum\limits_{s \in \Gamma } {f\left( s \right)g\left( {{s^{ - 1}}t} \right)} $$

for all t ∈Γ.