The Homology of Banach and Topological Algebras

One, Preparatory.- 0. Algebras, Modules, Complexes.- I. Cohomology Groups and Problems Giving Rise to Them.- II. Tensor Product.- Two, Basic.- III. Homological Concepts (General Properties).- IV. Projectivity.- V. Resolutions and Dimensions.- VI. Multi-Operational Holomorphic Calculus on the Taylor Spectrum.- VII. Flatness and Amenability.- Appendix A. Paracompact topological spaces.- Appendix B. Invariant means on locally compact groups.- Postscript.- §1. Extensions and derivations.- §2. Normal cohomology and its expression in terms of Ext.- §4. An interpretation of amenability-according-to-Connes in terms of the diagonal and reduced bifunctionals.- §5. “General homological” background to amenability according to Connes.- §6. Central contractibility (= central separability) and central cohomology.- §7. Homological dimensions. Results of a general character and results connected with the geometry of Banach spaces.- §8. Homological dimensions (continued). Algebras of smooth functions and some radical algebras.- §10. Homological dimensions (concluded). Connections with the question of an analytic structure on the spectrum.- §11. Miscellaneous results about the homological invariants of operator algebras and their modules.- §12. Completely bounded cohomology and its applications.- §13. Weakly amenable Banach algebras and various conditions for “ordinary” and weak amenability.- §15. Some remarks about the development (and metamorphosis) of the problems of a multi-operator holomorphic calculus.- References.- Postscript references.- Index of terminology.- Index of notation and abbreviations.