In this Research Note the authors exploit the relationship between algebra and spectral theory, in order to set into the context of Banach and C*-algebras the classical Riesz and Fredholm theory of bounded linear operators on Banach and Hubert spaces Fredholm theory is developed in primitive Banach algebras, and then extended to the general case, while Riesz theory follows as a consequence. Hubert space results are extended to the appropriate setting of C*-algebras, and numerous results are given. The book will be of particular interest to graduate students with a background in Banach algebras and spectral theory. Many of the results are new. and many are familiar results put into a new setting, the emphasis being on the underlying algebraic nature of classical spectral theory.
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