Proposition 7

The space A(G) is a self-adjoint ideal of $L^1(G)$. It is contained in $L^1(G)\cap L^2(G)$, and in the image of $\mathscr{C}(G)$ in $L^1(G)$.

Let $f\in L^1(G)$. For every $g\in L^2(G)$, one has $f*g\in L^1(G)$ (INT, VIII, §4, n$^o5$, prop. 12). Consequently, the space $L^1(G)\cap L^2(G)$ is an ideal of $L^1(G)$, and the same is true of the space A(G). The ideal A(G) is self-adjoint.

Let $f$ and $g$ be in $L^2(G)$. The convolution product $f*g$ is then the class of the continuous function given by

(INT, VIII, §4, n$^o5$, prop. 15). The second assertion follows.

Since $\chi (f*g) = (\chi f)*(\chi g)$ for $\chi \in \widehat{G},f\in L^1(G)$ and $g\in L^1(G)$ (INT, VIII, §3, No.$^o1$, prop. 6), one has $\chi h\in A(G)$ for every $h\in A(G)$ and $\chi \in \widehat{G}$. Since $\varepsilon_x*f=\boldsymbol{\gamma }(x)f$ and the linear representation $\boldsymbol{\gamma }$ is isometric on $L^p(G)$ for every $p$ (INT, VIII, §2, No.$^o5$, prop. 8), one has $\varepsilon_x*f\in A(G)$ for all $x\in G$ and $f\in A(G)$.

We denote by $\widehat{A}(G)$ the image of A(G) under the Fourier transform. It is a subspace of $\mathscr{C}_0(\widehat{G})$.