The theory of group representations plays an important roie in modern mathematics and its applica~ions to natural sciences. In the compulsory university curriculum it is included as a branch of algebra, dealing with representations of finite groups (see, for example, the textbook of A. I. Kostrikin ). The representation theory for compact, locally compact Abelian, and Lie groups is co- vered in graduate courses, concentrated around functional analysis. The author of the present boo~ has lectured for many years on functional analysis at Khar'kov University. He subsequently con- tinued these lectures in the form of a graduate course on the theory of group representations, in which special attention was devoted to a retrospective exposition of operator theory and harmo- nic analysis of functions from the standpoint of representation theory. In this approach it was natural to consider not only uni- tary, but also Banach representations, and not only representations of groups, but also of semigroups.
1 - Elements of Spectral Theory.- 1. Integration of vector-valued functions.- 2. Linear operators in Banach space.- 3. Spectrum and resolvent of linear operators.- 4. Invariant subspaces.- 5. Commutative Banach algebras.- 2 - Topological Groups and Semigroups.- 1. Topological groups.- 2. Topological semigroups.- 3. Invariant measures and means.- 3 - Elements of General Representation Theory.- 1. Actions and representations.- 2. Decomposition of representations.- 3. Finite dimensional representations.- 4. The representation spectrum of an Abelian semigroup.- 4 - Representations of Compact Semigroups.- 1. Harmonic analysis on compact groups.- 2. Banach representations of compact groups and semigroups.- 3. Almost periodic representations and functions.- 4. Nonnegative a.p. representations.- 5 - Representations of Locally Compact Abelian Groups.- 1. Elements of harmonic analysis.- 2. Representations with separable spectrum.- References.- Books.- Journal Articles.