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Riesz and Fredholm theory in Banach algebras. (English) Zbl 0534.46034
Research Notes in Mathematics, 67. Boston-London-Melbourne: Pitman Advanced Publishing Program. VI, 123 p. £7.95 (1982).
In this monograph the authors show how the classical theory of Fredholm and Riesz operators on Banach and Hilbert spaces can be extended to the setting of Banach and $$C^*$$-algebras.
There are six chapters. The first of these summarises results from Fredholm and Riesz operator theory, most of which are familiar. There follows a chapter on Fredholm theory in Banach algebras. The theory is initially developed in a unital primitive Banach algebra. A Fredholm element in such an algebra is defined as one which is invertible modulo the socle of the algebra. The concept is motivated by Atkinson’s characterization of a Fredholm operator on a Banach space as one which is invertible modulo the ideal of finite rank operators. The nullity, defect, and index of a Fredholm element can be defined, and a representation connects these quantities with the nullity, defect, and index of an associated Fredholm operator. This representation enables many results of Fredholm theory in primitive Banach algebras to be derived from their counterparts in operator theory. An extension to general unital Banach algebras is then possible by factoring out the primitive ideals. The numerical valued nullity, defect, and index in the primitive case are replaced by functions (with finite support) on the structure space of the general algebra. The classical spectral theory of Fredholm operators carries over in this general algebraic setting.
A development of Riesz theory in Banach algebras is the subject of the next chapter. A Riesz operator on a Banach space has been characterized by Ruston as one whose canonical image in the Calkin algebra has spectral radius zero. This characterization motivates the concept of a Riesz element x in a Banach algebra A, with respect to a closed two-sided ideal K, as one whose canonical image in the quotient algebra A/K has spectral radius zero. If K is an inessential ideal then, by building on the Fredholm theory developed in the previous chapter, a spectral theory for Riesz elements, analogous to that for Riesz operators, emerges for unital algebras. This is later extended to general Banach algebras. A theory of Riesz algebras is developed, and numerous examples of such algebras are given.
In the fourth chapter, results concerning Hilbert space operators are extended to a $$C^*$$-algebra setting. In particular the decomposition theorems of West and Stampfli are extended. It is shown that a $$C^*$$- algebra is a Riesz algebra if, and only if, the spectrum of every Hermitean element has no non-zero accumulation points.
A chapter is devoted to applications. Here the general theory is applied to the study of semi-normal elements in a $$C^*$$-algebra, to certain algebras of Banach space operators, as well as to measures on compact groups. A final chapter is intended to be used as a reference to that part of the general theory of Banach algebras which underlies the material in the text.
The subject matter is well presented. Each chapter concludes with a section of notes and comments which contain the historical development of the ideas discussed, and refer to an extensive bibliography. There are many new results, as well as old ones in a more general setting. The book will provide a welcome addition to the literature on Banach algebras and spectral theory.
Reviewer: H.P.Rogosinski

##### MSC:
 46H05 General theory of topological algebras 47A53 (Semi-) Fredholm operators; index theories 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 46L05 General theory of $$C^*$$-algebras 46J05 General theory of commutative topological algebras 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 47B40 Spectral operators, decomposable operators, well-bounded operators, etc.