Appears in collection : Introduction to Banach function algebras

In this course, we first give an introduction to the general theory of Banach algebras and a description of the most important examples including the operator algebra $B(E)$, the function algebra $C_0(X)$, the disk algebra $A(\Bbb D)$, the Lipschitz algebra, the algebra of continuously differentiable functions and the group algebra. Then an attempt is made to mention several basic results on Banach algebras. A key idea is that of the spectrum of an element which generalizes the concept of the eigenvalues of a matrix. Moreover, the maximal ideal space (the character space) of a Banach algebra is introduced to cover the Gelfand theory for commutative Banach algebras. Then we concentrate on semisimple commutative Banach algebras (the socalled Banach function algebras). We introduce some basic concepts in the theory of function algebras like the Choquet boundary and strong boundary points. The final part of this course is devoted to the study of some significant classes of operators to make a connection between the algebraic and topological structures of these Banach algebras.