Integrated forms are contractive nondegenerate star representations of L one

For $u,w\in C_c(G)$ one has $\pi (u\ast w)=\pi (u)\pi (w)$. Indeed, for $\xi ,\eta \in H$ the defining identity [F1] and the convolution formula give $⟨\pi (u\ast w)\xi ,\eta ⟩={\int }_G(u\ast w)(z)⟨\pi (z)\xi ,\eta ⟩dz={\int }_G{\int }_{G}u(x)w(x^{-1}z)⟨\pi (z)\xi ,\eta ⟩dxdz$; the integrand is continuous on the compact product $\mathrm{supp}u\times \{z:x^{-1}z\in \mathrm{supp}w\text{ for some }x\in \mathrm{supp}u\}$, so [F5] lets us substitute $z=xy$ (left invariance of Haar measure) and factor: ${\int }_G{\int }_{G}u(x)w(y)⟨\pi (x)\pi (y)\xi ,\eta ⟩dydx={\int }_{G}u(x)⟨\pi (x)\pi (w)\xi ,\eta ⟩dx=⟨\pi (u)\pi (w)\xi ,\eta ⟩$, using the defining weak integrals for $w$ and then $u$.

For $u\in C_c(G)$ one has $\pi (u^{\ast })=\pi (u{)}^{\ast }$. Indeed, for $\xi ,\eta \in H$ the defining identity and [F2] give $⟨\pi (u^{\ast })\xi ,\eta ⟩={\int }_G{\mathrm{\Delta }}_G(x^{-1})\overline{u(x^{-1})}⟨\pi (x)\xi ,\eta ⟩dx$; substituting $x=y^{-1}$ with [F3] and ${\mathrm{\Delta }}_G(x^{-1})dx=dy$ yields ${\int }_G\overline{u(y)}⟨\pi (y^{-1})\xi ,\eta ⟩dy={\int }_G\overline{u(y)}⟨\pi (y{)}^{\ast }\xi ,\eta ⟩dy=\overline{{\int }_{G}u(y)⟨\pi (y)\eta ,\xi ⟩dy}=\overline{⟨\pi (u)\eta ,\xi ⟩}=⟨\xi ,\pi (u)\eta ⟩=⟨\pi (u{)}^{\ast }\xi ,\eta ⟩$.

For every $\xi \in H$, $\pi (e_U)\xi \to \xi$: using $e_U\ge 0$, $\int e_U=1$ and the defining identity, $\mathrm{\parallel }\pi (e_U)\xi -\xi \mathrm{\parallel }\le {\int }_G{e}_U(g)\mathrm{\parallel }\pi (g)\xi -\xi \mathrm{\parallel }dg\le {\sup }_{g\in U}\mathrm{\parallel }\pi (g)\xi -\xi \mathrm{\parallel }$, which tends to $0$ along the directed set of identity neighbourhoods by strong continuity. Hence every $\xi$ lies in the closure of the span of $\{\pi (f){\xi }^{\mathrm{\prime }}\}$, and the closed span is $H$: it is a closed subspace containing every vector, so it is $H$, and no nonzero vector is annihilated by all $\pi (f)$; by [F6] this is nondegeneracy.

Multiplicativity for arbitrary $f,h\in L^1(G)$ follows from step 1.1 by density: for fixed $w\in C_c(G)$ both $f\mapsto \pi (f\ast w)$ and $f\mapsto \pi (f)\pi (w)$ are bounded complex-linear maps $L^1(G)\to \mathcal{B}(H)$, with bounds $\mathrm{\parallel }f\ast w{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }w{\mathrm{\parallel }}_1$ and $\mathrm{\parallel }\pi (f)\mathrm{\parallel }\mathrm{\parallel }\pi (w)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }\pi (w)\mathrm{\parallel }$, and they agree on the dense subspace $C_c(G)$; hence they agree for all $f$. Repeating with $f$ fixed and the variable $h$ — both sides bounded and linear in $h$ by [F1] and [F2], agreeing on $C_c(G)$ — gives $\pi (f\ast h)=\pi (f)\pi (h)$ for all $f,h\in L^1(G)$.

The involution identity extends by density: both $f\mapsto \pi (f^{\ast })$ and $f\mapsto \pi (f{)}^{\ast }$ are bounded conjugate-linear, hence continuous, maps $L^1(G)\to \mathcal{B}(H)$ (boundedness of the adjoint map uses $\mathrm{\parallel }{T}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$, available in the C*-algebra $\mathcal{B}(H)$), and they agree on the dense subspace $C_c(G)$ by step 1.2, hence everywhere.

By [F1] the map $f\mapsto \pi (f)$ is a bounded star-representation of the Banach $\ast$-algebra $L^1(G)$ with $\mathrm{\parallel }\pi (f)\mathrm{\parallel }\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$, by steps 2.1 and 2.2 it is multiplicative and star-preserving, and by step 1.3 it is nondegenerate; this is exactly the assertion that $f\mapsto \pi (f)$ is a nondegenerate star-representation of $L^1(G)$ in the sense of [F6], with contractive bound. The Axiom of Choice is inherited from the Haar, convolution, approximate-identity and Fubini suppliers of [F1]–[F5], and no further choice is used (The Axiom of Choice).