General theory of Banach algebras

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