Since the appearance of Gelfand's 1941 papers, there has been a rapid growth of interest in Banach algebras. The resulting development of the subject has brought the theory to a point where it is no longer just a promising tool in analysis but is an important field of research in its own right. Standing, as it were, between analysis and algebra (or perhaps more accurately, with feet in analysis and head in algebra), the theory of Banach algebras has developed roughly along two main lines representing respectively the analytic and algebraic influences. The analytic emphasis has been on the study of certain special Banach algebras, along with some generalizations of these algebras, and on extending certain portions of function theory and harmonic analysis to the more general situations offered by Banach algebras.
On the other hand, the algebraic emphasis has naturally been on various aspects of structure theory. Of great importance here has been the growing interest of algebraists in algebras without finiteness restrictions. This development, which has been much stimulated by the study of Banach algebras, has supplied important new algebraic methods which are profitably applied to Banach algebras. It becomes increasingly evident that, in spite of the deep and continuing influence of analysis on the theory of Banach algebras, the essence of the subject as an independent discipline is to be found in its algebraic development.
Author(s): C. E. Rickart
Edition: Revised
Publisher: Krieger
Year: 1974
Language: English
Pages: xvii+394
I. FUNDAMENTALS
Introduction.
§ 1. Definitions
§ 2. The regular representations.
§ 3. Complexiflcation of a real normed algebra
§ 4. The groups of regular and quasi-regular elements
§ 5. Topological divisors of zero
§ 6. The spectrum
§ 7. Normed division algebras
II. THE RADICAL; SEMI-SIMPLICITY AND THE 'STRUCTURE SPACES
Introduction
§ 1. Ideals and difference algebras
§ 2. Representations
§ 3. The radical.
§ 4. Primitive Banach algebras
§ 5. Uniqueness of the norm topology and the fundamental isomorphism theorem.
§ 6. Structure of semi-simple Banach algebras. The structure spaces
§ 7. Completely regular algebras.
§ 8. Annihilator algebras
III. COMMUTATIVE BANACH ALGEBRAS
Introduction.
§ 1. The carrier space and the Gelfand representation theorem
§ 2. Algebras of functions
§ 3. The Silov boundary.
§ 4. Representations of the carrier space
§ 5. Homomorphisms of certain function algebras into a Banach algebra.
§ 6. Direct-sum decompositions and related results
§ 7. Completely regular commutative Banach algebras
IV. ALGEBRAS WITH AN INVOLUTION
Introduction
§ 1. Miscellaneous properties of *-algebras
§ 2. Commutative *-algebras
§ 3. Self-dual vector spaces and *-representations
§ 4. Representations on Hilbert space
§ 5. Positive functionals and *-representations on Hilbert space
§ 6. Positive functionals and irreducible *-representations
§ 7. Symmetric *-algebras
§ 8. General properties of B*-algebras.
§ 9. Structure of ideals and representations of B*-algebras
§ 10. Banach *-algebras with minimal ideals.
APPENDIX EXAMPLES AND APPLICATIONS
Introduction.
§ 1. Algebras of operators
A.1.1. THE ALGEBRAS B(X) AND B(f).
A.1.2. THE ALGEBRA FG OF COMPACT OPERAT
A.1.3. THE SCHMIDT-CLASS FS
A.1.4. THE TxacE-CLass FT
A.1.5. W*-ALGEBRAS AND AW*-ALGEBR
A.1.6. SPECTRAL OPERATORS
§ 2. Algebras of functions.
A.2.1. THE ALGEBRA C(\Omega), FOR CERTAIN SPECIAL \Omega
A.2.2. THE l^p-ALGEBRS
A.2.3. FUNCTIONS WITH ABSOLUTELY CONVERGENT FOURIER SERIES
A.2.4. FUNCTIONS OF CLASS C^(n)
A.2.5. CONTINUOUS FUNCTIONS OF BOUNDED VARIATION
A.2.6. HOLOMORPHIC FUNCTIONS OF ONE VARIABLE.
A.2.7. HOLOMORPHIC FUNCTIONS OF SEVERAL VARIABLES.
A.2.8. A NON-SELF-ADJOINT ALGEBRA WITH 9ILOV BOUNDARY EQUAL TO THE CARRIER SPACE
A.2.9. NON-EXISTENCE OF THE SILOV BOUNDARY
A.2.10. ALGEBRAS OF SET FUNCTIONS ON THE LINE
A.2.11. SOME RADICAL ALGEBRAS
A.2.12. ALGEBRAS OF POWER SERIES
§ 3. Group algebras.
A.3.1. THE ALGEBRA L1(G)
A.3.2. LOCALLY COMPACT ABELIAN GROUPS
A.3.3. THE CONVOLUTION ALGEBRA OF MEASURES
A.3.4. GROUP ALGEBRAS OF A COMPACT GROUP
A. 3.5. ALMOST PERIODIC FUNCTIONS ON GROUPS.
A.3.6. GROUP ALGEBRAS OF OPERATORS
BIBLIOGRAPHY
LIST OF SYMBOLS
INDEX
