ABSTRACT
Chapter 6 is dedicated to the study of Banach algebras. The introduction of the concept of a Banach algebra is followed by a great number of examples and a study of the invertibility of elements in a complex Banach algebra with identity, and of the spectrum, resolvent and spectral radius. Also the concept of the topological divisor of zero is introduced and several important results related to this concept are established. Additional highlights include the study of subalgebras, one theorem by Hochwald–Morell as well as two theorems by Harte for regular elements in Banach algebras, conditions for the invertibility of operators in the Banach algebra of bounded linear operators from a Banach space into itself. Also detailed studies of the spectra of adjoint, normal and compact operators are included. Finally, the concept of C ∗ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429442599/3ba74af1-90b9-447a-a656-b532a039cc7a/content/inline-math6_1.jpg"/> –algebras is introduced, their most important properties are established and conditions for the invertibility of the difference of projections are given.
