L1 of a locally compact group is a Banach star-algebra

Associativity of convolution on $L^1(G)$. Let $F,G\in L^1(G)$ and $h\in C_c(G)$, and choose $f_n,g_n\in C_c(G)$ with $f_n\to F$ and $g_n\to G$ in $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$, possible by [F5] under the AC of [A1]. By [F2] the products converge: $f_n\ast g_n\to F\ast G$ and $g_n\ast h\to G\ast h$. Applying [F2] again, $(f_n\ast g_n)\ast h\to (F\ast G)\ast h$ and $f_n\ast (g_n\ast h)\to F\ast (G\ast h)$. By [F3] the two sequences are equal termwise, so their limits are equal: $(F\ast G)\ast h=F\ast (G\ast h)$ for $F,G\in L^1(G)$ and $h\in C_c(G)$.

Associativity on $L^1(G)$ in the second variable as well. Let $F,G,H\in L^1(G)$ and choose $h_n\in C_c(G)$ with $h_n\to H$. By step 1.1, $(F\ast G)\ast h_n=F\ast (G\ast h_n)$ for every $n$. By joint continuity [F2], $(F\ast G)\ast h_n\to (F\ast G)\ast H$ and $G\ast h_n\to G\ast H$, hence $F\ast (G\ast h_n)\to F\ast (G\ast H)$; uniqueness of limits gives $(F\ast G)\ast H=F\ast (G\ast H)$.

The structural axioms hold. $L^1(G)$ is a complex vector space, complete and normed, with associative bilinear multiplication that is submultiplicative by [F2] and associative by step 2.1; the involution is conjugate-linear and involutive and reverses products by [F4], and it is isometric, hence in particular continuous. This is exactly the list of properties required of a complex Banach $\ast$-algebra without a required unit in [F6], and no unit is claimed to exist. ∎