A sequential approximate identity concentrated near the identity

By [F6], list the basis members containing $e$ as $(B_k{)}_{k\in \mathbb{N}}$, repeating members if there are only finitely many, and put $V_n={\bigcap }_{k=0}^n{B}_k$. Each $V_n$ is an identity neighbourhood, $V_{n+1}\subseteq V_n$, and for every identity neighbourhood $U$ there is $N$ such that $V_n\subseteq U$ for all $n\ge N$: choose a basis member $B_j$ with $e\in B_j\subseteq U$ and take $N=j$. Define $u_n=e_{V_n}$ using the net in [F3]. By [F4], each such compactly supported function defines an $L^1(G)$ class, and its nonnegativity gives ${\int }_G{u}_{n}d\mu =\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_1$. Thus $u_n\in C_c(G)$, $u_n\ge 0$, $\mathrm{supp}{u}_n\subseteq V_n$, and ${\int }_G{u}_{n}d\mu =\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_1=1$. The countability lemma supplies the enumeration without choice; AC is inherited from the net supplier [F3].

Fix $f\in L^1(G)$ and $ϵ>0$. By [F3], there is an identity neighbourhood $W$ such that both $\mathrm{\parallel }{e}_U\ast f-f{\mathrm{\parallel }}_1<ϵ$ and $\mathrm{\parallel }f\ast e_U-f{\mathrm{\parallel }}_1<ϵ$ whenever $U\subseteq W$. By step 1.1 choose $N$ with $V_N\subseteq W$. For every $n\ge N$, $V_n\subseteq V_N\subseteq W$, so $u_n=e_{V_n}$ satisfies both inequalities. This proves the two stated $L^1$ limits.

Let $C_{\rho }$ be a bound for $\rho$ from [F8]. The set $q(C_c(G))$ is dense in $C^{\ast }(G)$: approximate first by $q(f)$ with $f\in L^1(G)$ using [F2], then approximate $f$ in $L^1$ by a member of $C_c(G)$ using [F1] and apply $\mathrm{\parallel }q(g){\mathrm{\parallel }}_{C^{\ast }}\le \mathrm{\parallel }g{\mathrm{\parallel }}_1$ from [F9]. For $a\in C^{\ast }(G)$ and $\eta \in H$, boundedness of $\rho$ carries approximations $q(c)\to a$ to $\rho (q(c))\eta \to \rho (a)\eta$; thus nondegeneracy [F8] makes the linear span of $\rho (q(c))\eta$ dense in $H$. For each $c\in C_c(G)$ and $\eta \in H$, [F5] gives $u_n\ast c\in C_c(G)$, and the star-homomorphism identity for $q$ gives $\rho (q(u_n))\rho (q(c))\eta -\rho (q(c))\eta =\rho (q(u_n\ast c-c))\eta$. Its norm is at most $C_{\rho }\mathrm{\parallel }q(u_n\ast c-c){\mathrm{\parallel }}_{C^{\ast }}\mathrm{\parallel }\eta \mathrm{\parallel }\le C_{\rho }\mathrm{\parallel }{u}_n\ast c-c{\mathrm{\parallel }}_1\mathrm{\parallel }\eta \mathrm{\parallel }$, which tends to zero by step 2.1. Linearity gives convergence on finite linear combinations of these vectors. Moreover, $\mathrm{\parallel }\rho (q(u_n))\mathrm{\parallel }\le C_{\rho }\mathrm{\parallel }q(u_n){\mathrm{\parallel }}_{C^{\ast }}\le C_{\rho }\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_1=C_{\rho }$, uniformly in $n$. For any $\xi \in H$ and any ${\xi }_0$ in that dense span, $\mathrm{\parallel }\rho (q(u_n))\xi -\xi \mathrm{\parallel }\le (C_{\rho }+1)\mathrm{\parallel }\xi -{\xi }_0\mathrm{\parallel }+\mathrm{\parallel }\rho (q(u_n)){\xi }_0-{\xi }_0\mathrm{\parallel }$; density and convergence on the span therefore give convergence for every $\xi$. If $H=\{0\}$ the strong limit statement is immediate.