The Homology of Banach and Topological Algebras

One, Preparatory.- 0. Algebras, Modules, Complexes.- x1. Banach and locally convex algebras. Indispensable concepts and facts.- 1.1. Minimum background in pure algebra (theory of associative algebras).- 1.2. Minimum background in the theory of locally convex spaces. Vector-valued analytic functions.- 1.3. General locally convex algebras.- 1.4. Banach algebras.- x2. Banach and locally convex algebras. Indispensable examples.- 2.1. Banach function algebras and Banach sequence algebras.- 2.2. Group algebras.- 2.3. Operator algebras.- 2.4. The algebra of holomorphic functions on a domain and other non-normed algebras.- x3. Modules (representations).- 3.1. Algebraic modules.- 3.2. Locally convex modules. Concepts and facts.- 3.3. Locally convex modules. Examples.- x4. Categories of modules and their associated functors.- 4.1. The background in category theory. Standard categories of Banach and locally convex modules.- 4.2. The forgetful, unitization and replacement functors. The morphism functor "Ah" and its analogues.- x5. Complexes and the homology functor.- 5.1. Exact sequences.- 5.2. The case of Banach modules: a theorem on the relation between the exactness of a sequence and the exactness of its dual.- 5.3. Complexes and the homology functor.- 5.4. The "fundamental lemma of homological algebra" and conditions for a given algebraic isomorphism to be topological.- I. Cohomology Groups and Problems Giving Rise to Them.- x1. Extensions.- 1.1. General concepts.- 1.2. Singular extensions and the space H2(A, X).- 1.3. Annihilator and finite-dimensional extensions; connection with the geometry of the unit ball.- x2. Derivations and other questions.- 2.1. Derivations and the space H1(A, X).- 2.2. Perturbation of algebras and modules. The space H3(A, X).- x3. Standard complexes and cohomology groups.- 3.1. Definitions and the basic questions.- 3.2. Some remarks on "direct" methods.- Notes.- II. Tensor Product.- x1. Introductory concepts.- 1.1. Universality property. Algebraic tensor product.- 1.2. Tensor products of seminorms.- x2. The tensor product of Banach spaces.- 2.1. Definition and explicit construction.- 2.2. Examples. Tensor multiplication on L1(u), C(?) and Hilbert space.- 2.3. The tensor product of operators and the functor "$$hat{ otimes }$$".- 2.4. The tensor product of spaces in a dual pair. Nuclear operators. The numerical and operator trace.- 2.5. Approximation property. Application to the problem of the existence of a trace.- 2.6. Bounds for norms of diagonal and triangular elements.- 2.7. The weak tensor product and other kinds of tensor product.- x3. The tensor product of Banach modules.- 3.1. Definition and general properties.- 3.2. Tensor multiplication by ideals and cyclic modules.- 3.3. Applications to annihilator extensions.- x4. Topological tensor products.- 4.1. The projective and inductive tensor product.- 4.2. Tensor multiplication by an algebra of holomorphic functions and other examples.- x5. Algebras, modules and complexes revisited (additional material based on the tensor product).- 5.1. Tensor product of algebras.- 5.2. The enveloping algebra and the reduction of all modules to left unital modules.- 5.3. The functor "$$ mathop{ otimes }limits_A^{ wedge } $$" and its properties. Conjugate associativity.- 5.4. Bicomplexes and the tensor product of complexes.- 5.5. Homology groups.- Notes.- Two, Basic.- III. Homological Concepts (General Properties).- x1. Projective Banach and locally convex modules.- 1.1. Homotopy and the splitting of complexes.- 1.2. Projective and injective modules.- 1.3. Free modules. Lifting problems characterizing projectivity. Free modules over O(U).- 1.4. Co-free Banach modules and their relation with injective modules. Non-unital projective modules and bimodules.- x2. Resolutions.- 2.1. Projective resolutions and the comparison theorem.- 2.2. Normalized bar-resolution.- 2.3. Non-normalized bar-resolution. Versions of the standard resolutions for non-unital modules and bimodules.- x3. Derived functors.- 3.1. Definition of derived functors and the long exact sequence.- 3.2. Independence of the choice of resolutions.- x4. Ext and Tor.- 4.1. Ext and its connection with lifting and extension problems.- 4.2. Expressing cohomology groups in terms of Ext.- 4.3. Applications: cohomology and derivations of operator algebras.- 4.4. Tor and its connection with homology groups.- x5. Homological dimensions of modules and algebras.- Notes.- IV. Projectivity.- x1. Some general methods by which projectivity may be checked.- x2. Projectivity of ideals; sufficient conditions.- 2.1. Projective and hereditary algebras.- 2.2. Canonical projections for ideals of function algebras and group algebras. Projectivity in terms of an operator extending functions from the diagonal.- 2.3. Ideals in C(?); the role of the topological properties of the spectrum.- 2.4. Ideals in operator and group algebras.- x3. Intrinsic (necessary) properties of projective ideals.- 3.1. The skeleton of a projective ideal.- 3.2. Application: the paracompactness of spectra and a description of the projective ideals in C(?).- 3.3. Stability-type conditions.- 3.4. Elements with not very large norms.- x4. Algebras with projective cyclic modules.- 4.1. Posing the basic questions.- 4.2. Realization of cyclic modules as ideals; the role of the approximation property.- 4.3. Algebras of global dimension zero.- x5. Biprojective algebras.- 5.1. Definition and general properties. The retraction problem characterizing biprojectivity.- 5.2. Examples. Biprojective algebras among group and operator algebras.- 5.3. The structure of biprojective algebras; conditions for them to be representable as direct sums of algebras of nuclear operators.- 5.4. Non-normed algebras of bidimension zero; characterizations of ?M.- Notes.- V. Resolutions and Dimensions.- x1. Koszul resolution.- 1.1. Koszul complex.- 1.2. Koszul resolution for algebras of holomorphic functions.- x2. The entwining resolution and dimensions of Banach algebras.- 2.1. Entwining resolution.- 2.2. Morphisms that do not extend from the diagonal.- 2.3. Lower bounds for the global dimension of function algebras. Applications to singular extensions.- 2.4. Dimensions of biprojective algebras.- 2.5. Unsolved problems and miscellaneous remarks.- Notes.- VI. Multi-Operational Holomorphic Calculus on the Taylor Spectrum.- x1. The Taylor spectrum and the formulation of the basic theorem.- x2. The complex dominating a module.- x3. Exact complexes of spaces of holomorphic functions. The connection between exactness on the fibres and global exactness.- x4. Construction of the dominating ?ech complex - end of the proof of the basic theorem.- Notes.- VII. Flatness and Amenability.- x1. Flat modules.- 1.1. Definition of a flat module. A sufficient condition for an ideal to be flat.- 1.2. Comparison of ToroA(X, Y) and $$ Xmathop{ otimes }limits_A^{ wedge } Y $$. Non-flat modules with the Tor of positive dimension all trivial.- 1.3. Interrelation between flatness and injectivity.- 1.4. A criterion for cyclic modules to be flat.- x2. Amenable algebras.- 2.1. Injective bimodules and biflat algebras. The coretraction problem characterizing biflatness.- 2.2. The diagonal ideal of an enveloping algebra.- 2.3. Amenable algebras and their equivalent definitions.- 2.4. Certain properties of amenable algebras.- 2.5. Amenable group algebras and Johnson's theorem.- 2.6. Amenable uniform algebras; a characterization of C(?). Some remarks about amenable C?-algebras.- Notes.- Appendix A. Paracompact topological spaces.- Appendix B. Invariant means on locally compact groups.- Postscript.- x1. Extensions and derivations.- x2. Normal cohomology and its expression in terms of Ext.- x4. An interpretation of amenability-according-to-Connes in terms of the diagonal and reduced bifunctionals.- x5. "General homological" background to amenability according to Connes.- x6. Central contractibility (= central separability) and central cohomology.- x7. Homological dimensions. Results of a general character and results connected with the geometry of Banach spaces.- x8. Homological dimensions (continued). Algebras of smooth functions and some radical algebras.- x10. Homological dimensions (concluded). Connections with the question of an analytic structure on the spectrum.- x11. Miscellaneous results about the homological invariants of operator algebras and their modules.- x12. Completely bounded cohomology and its applications.- x13. Weakly amenable Banach algebras and various conditions for "ordinary" and weak amenability.- x15. 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.