Gel'fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis

We define a general notion of Gel'fand-Tsetlin algebras in the framework of inductive limit of the family semisimple star-algebras and the notion of genralized conditional expectations. The inmverse limit of such family with respect to generalized expectation is bimodule over inductive limit. The most important examples conserned to inductive families of group algebras of classical series of the discrete and continuous groups. I start from some remarks about distinguished role of I.M.Gel'fand's conceptions in whole area.