On the mathematical work of Hans Reiter
The main studies of Hans Reiter, belonging to his impressive mathematical work, discussed by the author, are: ``Investigations in harmonic analysis'', demonstrating the theorem of Wiener-Lévy for locally compact Abelian groups and investigating the structure of the quotient \(L^ 1(G)/I\) for a closed ideal \(I\) of \(L^ 1(G)\); ``On a certain class of algebra ideals in the \(L^ 1\)-algebra of locally compact Abelian group'', proving that if the cospectrum of \(I\) is a countable independent subset \(S\) and \(\widehat G\), then \(L^ 1(G)/I\) is isomorphic to the Banach algebra of all continuous functions vanishing at infinity on \(S\); ``Contributions to harmonic analysis, II, III'', containing one of the most beautiful results of his whole work; ``The convex hull of translates of a function in \(L^ 1\)'', giving an elegant formula for the distance between 0 and the convex set generated by \(f_ h\) \(h\) \(H\), where \(H\) is an arbitrary Abelian closed subgroup of \(G\); ``Classical harmonic analysis and locally compact groups'', his famous book, published in 1968 (see Zbl 0165.156), based on a principle enunciated by A. Weil, contains most of the subjects treated in his studies (referring to Bourbaki's integration theorem, to Schwartz's counterexample to spectral synthesis, to C. Herz's proof of the spectral synthesis). Worth mentioning are both the formulations of some general principles, such as the principle of relativization, and some valuable proofs on various mathematical topics. The book, a real ``gem'', opens new directions of research, mainly in the context of non-commutative groups. Other contributions of Reiter's discussed here are: ``Sur certain idéaux dans \(L^ 1(G)\)''; ``\(L^ 1\) algebras and Segal algebras''; ``Über den Satz von Wiener und lokalkompakte Gruppen''.
scientific article; zbMATH DE number 164904
scientific article