Definition 44.2.7.label Let $G$ be a group. For each $x \in G$, let $u_{x} = \lambda(x)$ and identify it with $\delta_{x}$, then $\lambda(G) = \bracsn{u_x|x \in G}$ are the canonical unitaries of $L(G)$.

For each $f \in l^{2}(G)$, $f = \sum_{x \in G}f(x)u_{x}$. In particular, for each $T \in L(G) = LC(G) \subset l^{2}(G; \complex)$, there exists $\bracsn{T_x}_{x \in G}\subset \complex$ such that $T$ may be written as $T = \sum_{x \in G}T_{x} u_{x}$^[1]. In which case, the scalars $\bracsn{T_x}_{x \in G}$ are the Fourier coefficients of $T$.