Espaces symétriques ordonnés et algèbres de Volterra. (Ordered symmetric spaces and Volterra algebras)
Let \(X=G/H\) be a symmetric space and S a subsemigroup of G which contains H as the largest subgroup. Then S/H forms the positive cone of a left invariant partial ordering on X. A causal kernel on X is a function F: \(X\times X\to {\mathbb{C}}\) which is supported on the graph of the ordering. If the order intervals \(\{\) z: \(x\leq z\leq y\}\) are compact one can form a convolution product of causal kernels and in this way obtain the Volterra algebra, which plays an important role in the harmonic analysis of the space X. The paper under review gives an introduction to the theory and describes a class of symmetric spaces of type \(G_{{\mathbb{C}}}/G_{{\mathbb{R}}}\) to which it applies. Finally the compactness of order intervals is shown for this class.
scientific article; zbMATH DE number 4210800