Closed ideals in convolution algebras and the Laplace transform
The convolution algebra \(L^ 1([0,1]^ n)\) on the n-fold product of the unit interval has an easily identified ideal structure when \(n=1\); the closed ideals are of the form \(M_{\beta}:=\{f:\) inf(ess.supp(f))\(\geq \beta \}\). This paper shows that this is not the case when \(n>1\). It then proceeds to establish a correspondence between the closed ideals of \(L^ 1([0,1]^ n)\) and the closed ideals in the quotient \(A_ 0^{(n)}/K\) of the algebra \(A_ 0^{(n)}\) consisting of the functions of n complex variables, continuous on the Cartesian product of n copies of the closed right half-plane, analytic on the interior of this set, vanishing at \(\infty\); K is an ideal generated by specified exponentials. The map establishing the correspondence is (essentially) the inverse Laplace transform.
scientific article; zbMATH DE number 4120813