Topological Algebras in the Theory of Generalized Functions

The goal of the proposed research project is to substantially progress the structure of algebras of Colombeau generalized functions as topological algebras over the ring of generalized constants C; and to develop a theory of Colombeau Banach algebras. This is a continuation of recent research on Colombeau generalized functions as topological modules over C. More specifically, the classical theory of topological algebras (over the field of complex constants) contains central results whose generalization will yield new insight into the structure of Colombeau generalized functions: for example, the so-called Gel`fand theory shows that a wide class of commutative topological algebras (including any C*-algebra) can be represented as an algebra of continuous functions on a compact Hausdorff topological space. In the case of an algebra of generalized functions on a compact set K, this space is at least as rich as the space of generalized points of K - a space on which Colombeau generalized functions already have an interpretation as pointwise functions. In the same way, and as evidenced by the research on topological modules over C, it will be possible to develop counterparts and generalizations of further classical concepts from topological rings and algebras in the Colombeau setting, thereby considerably enlarging the domain of applicability, e.g., in operator theory. The fact that C is not a field will affect the role of the spectral theory played in Gel`fand theory and produce new interesting structures. In this respect, useful insight is expected from theories of Colombeau generalized functions within nonstandard analysis, in which the ring of generalized constants remains a field. In this way, the research will yield a cross-fertilization of ideas from which mainly the theories of generalized functions, topological ring theory, Gel`fand theory, operator theory and C*-theory (with its applications to quantum mechanics) will benefit.