General Theory of Hypercomplex Systems

Abstract

In 1950, when extending the Delsarte-Levitan theory of generalized translation operators, Yu. M. Berezanskii and S. G. Krein introduced the concept of (commutative) hypercomplex system with continuous basis and developed harmonic analysis for such systems. Each hypercomplex system is a Banach *-algebra of functions on a locally compact space (the basis of a hypercomplex system). It generalizes the concept of hypercomplex system with finite basis and the concept of locally compact group algebra. The role of the group translation operators is played by generalized translation operators, which are naturally associated with a hypercomplex system. Note that it is possible to completely characterize hypercomplex systems in terms of such generalized translation operators (Theorem 2.1) and, hence, a hypercomplex system with continuous basis can be considered as a class of generalized translation operators that admit the construction of rich harmonic analysis and duality theory.