Some applications of Boolean valued set theory to abstract harmonic analysis on locally compact groups

Hirokazu Nishimura
1985 Publications of the Research Institute for Mathematical Sciences  
The main purpose of this paper is to extend Takeuti's [23] Boolean valued treatment of abstract harmonic analysis on locally compact abelian groups to locally compact groups (neither abelian nor compact in general). The distinctive feature of our approach, compared with traditional treatments of the subject, is that we can establish many important theorems without resort to direct integrals or to the theory of Banach algebras. By way of illustration, we will give such a proof of renowned
more » ... of renowned Bochner's theorem. This paper is not intended to be exhaustive at all but hopefully to be suggestive. How far we can proceed in this direction yet remains to be seen. Abstract harmonic analysis has two origins. One is the classical Fourier analysis set forth, e.g., in Bochner [3] and Zygmund [31] . The other is the algebraic theory of finite groups and their representations, whose modern and comprehensive treatment can be seen, e.g., in Curtis and Reiner [5] . Indeed the spirit of abstract harmonic analysis is to do Fourier analysis on topological groups as general as possible, guided by the representation theory of finite groups while using the modern techniques of functional analysis. The most central technique in the study of topological groups, which are usually assumed to be locally compact at least, is their unitary representations (on some appropriate Hilbert spaces) and, in particular, their irreducible unitary representations. As for compact groups, it is well known that their irreducible unitary representations are finite-dimensional and any unitary representation of such a
doi:10.2977/prims/1195179842 fatcat:klndzjkkbbaexoefnumifa3pia