Exercise 17, § 3

For every $p\geqslant 1$, we identify $L^p(\mathbf{R}_+)$ with a closed subspace of $L^p(\mathbf{R})$.

Let $f\in L^1(\mathbf{R}_+)$ be the function defined by $f(x) = 2e^{-x}$ for $x\in \mathbf{R}_+$, and let $g\in L^1(\mathbf{R})$ be the function such that $g(x) =f(-x)$.

a) Now prove that $f+g=f*g$.

b) Let A (resp. B) be the subalgebra of $L^1(\mathbf{R})$ generated by $f$ (resp. by $f$ and $g$). Now prove that A is dense in $L^1(\mathbf{R}_+)$. (Let $h\in L^{\infty}(\mathbf{R}_+)$ such that $\langle h, f^{*n}\rangle = 0$ for every integer $n\geqslant 1$, where $f^{*n}$ is the convolution iterated $n$ times of $f$; now prove that the Laplace transform

defined for every $z\in \mathbf{C}$ such that $\mathscr{R}(z)>0$, is zero.) Deduce that B is dense in $L^1(\mathbf{R})$.

c) Let C be a closed proper subalgebra of $L^1(\mathbf{R})$ containing A. One has $1\in$ Sp$_C(f)$. (Prove that otherwise, question a) and the holomorphic functional calculus in C would imply that $g\in C$.) d) There exists a character $\chi$ of C such that $\chi (f) = 1$; one has

for every function $\varphi \in L^1(\mathbf{R}_+)$.

e) There exists a measure $\mu\in \mathscr{M}^1(\mathbf{R})$ of norm 1 such that $\chi (\varphi ) =\int\widehat{\varphi d\mu}$ for every function $\varphi \in C$.

f) The Fourier transform of $\mu$ is the function $x\mapsto e^{-|x|}$.

g) Let $\varphi \in C$. For every $y >0$, one has

(Use the relation $\chi (\varphi *\psi ) =\chi (\varphi )\chi (\psi )$ for $\psi \in L^1(\mathbf{R}_+)$.)

h) One has $C = L^1(\mathbf{R}_+)$. (Thus, the subalgebra $L^1(\mathbf{R}_+)$ is a maximal closed subalgebra of $L^1(\mathbf{R})$.)