Introduction to Banach Algebras, Operators, and Harmonic Analysis
The book contains a series of five lectures for graduate students given in two conferences in 1999 and 2002. The first lecture by H. G. Dales is an introduction to the theory of Banach algebras. It contains the Gelfand theory, the functional calculus and properties of homomorphisms and derivations. In the last part of this lecture, a relation of cohomology theory to amenability is described. The last topic is then investigated in connection with group algebras L1(G) in the second lecture, written by G. A. Willis. Automatic continuity of derivations from L1(G) is shown for several special cases of G at the end of this lecture. The second part of the book is devoted to theory of B-algebras of continuous linear operators. The central topic here is the invariant subspace problem. The third lecture by J. Eschmeier develops a method due to S. Brown, which is based on H∞ functional calculus. This method leads to the proof of existence of an invariant subspace for several classes of operators (contractions with dominating spectra and subnormal or, more generally, subdecomposable and reflexive operators). The local spectral theory is investigated in the fourth lecture by K. B. Laursen for a class of operators, which includes decomposable operators. The duality between the so-called β and δ properties is described here together with an extension of the notion of similarity of operators. This part ends with an application of local spectral theory to multipliers on a commutative B-algebra. The last lecture by P. Aiena is closely related to the previous one. It studies properties of decomposable operators, which are similar to properties of compact and normal operators. The so-called single-valued extension property (SVEP) plays an important role in a generalization of Fredholm operators. The reader of the book can benefit from valuable comments on further results and relations, which are attached to each section. There are many exercises at the end of each section emphasizing a didactic character of the lecture notes.