Cyclic unitary representation of a positive-definite function ($p(x)=⟨U_{x}v,v⟩$)

For an Abelian group, on finite formal sums of group points, define $⟨[x],[y]⟩=p(x{y}^{-1})$, with the first variable linear. Quotient by the null space of this positive semidefinite form and complete to a Hilbert space. Left translation is unitary, and $v=[e]$ generates a dense span of translated vectors. For continuous $p$, the equality $\Vert U_x[y]-[y]{\Vert }^2=2(p(e)-\mathrm{Re}p(x))$ gives strong continuity. This realizes $p$ as a matrix coefficient without assuming its integral representation.