Gelfand-Tsetlin algebras, expectations, inverse limits, Fourier analysis

Abstract

This text mainly follows my talk at the conference “Unity of Mathematics” (Harvard, September 2003), devoted to the 90th birthday of I. M. Gelfand. I introduce some new notions that are related to several old ideas of I. M. and try to give a draft of the future development of this area, which includes the representation theory of inductive families of groups and algebras and Fourier analysis on such groups. I also include a few reminiscences about I. M. as my guide.