THE DAUNS–HOFMANN THEOREM REVISITED

Abstract

John Dauns died on June 4, 2009 of cancer in New Orleans at the age of 73. His work on rings and modules is well-known in the algebra community. However, functional analysts working in the area of C*-algebras are likely to know his name from one theorem that is a corollary of results he and I had obtained in work we did in the mid-sixties of the last century ([17, 27, 18]) and which became known as the Dauns-Hofmann Theorem in C*-algebra theory. It has been known in the C*-text book and monograph literature up to the recent one under this name ([9, 20, 21, 39, 41]) since it provides apparently a useful tool that continues to be used in current research (see e.g. [7, 8, 25, 38, 40]). The problem with the historical record of the Dauns–Hofmann Theorem is that it used to be somewhat obscure how it originated and that the full weight of what was proved was not precisely understood for a long time. As John Dauns was deeply, if not subbornly involved in the development of the early phases of the representation of rings, algebras, C*-algebras (and other classes of algebras) by continuous sections in bundles (sometimes called continuous fields) [11–18], and since his work in this area was substantial and contributed much to a local culture of "sectional representation" at Tulane University ([19, 22, 27–34, 43, 44, 48]), I feel that it is justified to attempt a clarification. He can no longer participate himself in such an attempt; nor would he actually protest the occasional lack of acknowledgment were he alive, because that would be contrary to his ever gentle disposition. This small survey is devoted to shedding some additional light on this portion of John Dauns' work in mathematics; it is natural that it should have a personal tenor by a writer remembering his presence and his work as a collaborator.