Convolution on L1 of a locally compact group

Assume AC. Let $G$ be an LCH group with a fixed left Haar measure $\mu$, write $C_c(G):=C_c(G;\mathbb{C})$ for the continuous complex-valued functions of compact support and $L^1(G):=L^1(G,\mu ;\mathbb{C})$ for the complex Haar space with its norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ (Complex Haar L^p spaces and compactly supported functions), and let the symbol $\ast$ denote the convolution of $C_c(G)$ (Compactly supported convolution on a group).

The extension. There is exactly one map $L^1(G)\times L^1(G)\to L^1(G)$, again written $(f,g)\mapsto f\ast g$ and called convolution on $L^1(G)$, with all three of the following properties.

1. It is $\mathbb{C}$-bilinear: $(\alpha f+\beta f^{\mathrm{\prime }})\ast g=\alpha (f\ast g)+\beta (f^{\mathrm{\prime }}\ast g)$ and $f\ast (\alpha g+\beta g^{\mathrm{\prime }})=\alpha (f\ast g)+\beta (f\ast g^{\mathrm{\prime }})$ for all $\alpha ,\beta \in \mathbb{C}$ and $f,f^{\mathrm{\prime }},g,g^{\mathrm{\prime }}\in L^1(G)$.
2. It is bounded, hence jointly continuous, with $\mathrm{\parallel }f\ast g{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1(f,g\in L^1(G));$ consequently the map $(f,g)\mapsto f\ast g$ is continuous for the product of the norm topologies.
3. It agrees with the $C_c$ convolution of Compactly supported convolution on a group whenever both arguments lie in $C_c(G)$.

Representatives are not part of the data. For a general $f\in L^1(G)$ the symbol $f(x)$ has no meaning: $f$ is an equivalence class, and no pointwise formula is asserted for the extended product. What is asserted is that for $f\in L^1(G)$ and $g\in C_c(G)$ the class $f\ast g$ is the limit in $L^1(G)$ of $u_n\ast g$ for any sequence $u_n\in C_c(G)$ with $u_n\to f$, and dually in the second variable; the $\mu$-a.e. integral formula $g\mapsto {\int }_{G}f(y)g(y^{-1}x)d\mu (y)$ for the representative of $f\ast g$ is not claimed here and is not used on this page except through the class-level identity $f\ast g$ for $C_c$ arguments.

Well-definedness. Existence. Fix $g\in C_c(G)$. If $u_n\in C_c(G)$ converge to $f$ in $L^1(G)$, then $u_n\ast g$ is a Cauchy sequence in $L^1(G)$, because $\mathrm{\parallel }{u}_n\ast g-u_m\ast g{\mathrm{\parallel }}_1=\mathrm{\parallel }(u_n-u_m)\ast g{\mathrm{\parallel }}_1\le \mathrm{\parallel }{u}_n-u_m{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1$ by the $C_c$ norm inequality (Submultiplicativity of convolution in the L1 norm), and $L^1(G)$ is complete (Completeness of the complex Haar L1 and L2 spaces and density of Cc); such a sequence $u_n$ exists because $C_c(G)$ is dense in $L^1(G)$ (same item). The limit is independent of the choice of $(u_n)$, because the interleaving of two such sequences is again a sequence in $C_c(G)$ converging to $f$, so both limits equal its limit. Passing to the limit in the inequality for the approximants gives $\mathrm{\parallel }f\ast g{\mathrm{\parallel }}_1\le \mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1$, and the assignment $f\mapsto f\ast g$ is linear on $C_c(G)$ and bounded, so it extends to a linear map on $L^1(G)$ with the same bound. Repeating the construction in the second variable produces the two-variable map, and the two constructions agree on $C_c(G)\times C_c(G)$: for $f,g\in C_c(G)$ every approximant may be taken equal to $f$ or to $g$, and the two-order computation gives the same limit because $\mathrm{\parallel }{u}_n\ast g-f\ast v_m{\mathrm{\parallel }}_1\le \mathrm{\parallel }{u}_n-f{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1+\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }g-v_m{\mathrm{\parallel }}_1\to 0$.

Bilinearity is inherited from the approximants: for $f,f^{\mathrm{\prime }}\in L^1(G)$, scalars $\alpha ,\beta$ and $g\in C_c(G)$, approximating $f$ and $f^{\mathrm{\prime }}$ by $u_n,u_n^{\mathrm{\prime }}\in C_c(G)$ gives approximants $\alpha u_n+\beta u_n^{\mathrm{\prime }}$ of $\alpha f+\beta f^{\mathrm{\prime }}$, and $(\alpha u_n+\beta u_n^{\mathrm{\prime }})\ast g=\alpha (u_n\ast g)+\beta (u_n^{\mathrm{\prime }}\ast g)$ by bilinearity on $C_c(G)$; uniqueness of limits in $L^1(G)$ gives $(\alpha f+\beta f^{\mathrm{\prime }})\ast g=\alpha (f\ast g)+\beta (f^{\mathrm{\prime }}\ast g)$, and a second approximation in the variable $g$ gives the same identity for general $g\in L^1(G)$.

Uniqueness. The subset $C_c(G)\times C_c(G)$ is dense in $L^1(G)\times L^1(G)$: given $f,g$ and $ϵ>0$, density of $C_c(G)$ provides $u,v\in C_c(G)$ with $\mathrm{\parallel }u-f{\mathrm{\parallel }}_1<ϵ$ and $\mathrm{\parallel }v-g{\mathrm{\parallel }}_1<ϵ$, so $(u,v)$ is within $ϵ$ of $(f,g)$ for the product metric. Any two maps $L^1(G)\times L^1(G)\to L^1(G)$ satisfying properties 1–3 are continuous by 2 and agree on that dense subset, hence they agree everywhere: a norm-continuous map on a metric space is determined by its values on a dense subset.