The Homology of Banach and Topological Algebras [Paperback]

One, Preparatory.- 0. Algebras, Modules, Complexes.- ?1. 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.- ?2. 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.- ?3. Modules (representations).- 3.1. Algebraic modules.- 3.2. Locally convex modules. Concepts and facts.- 3.3. Locally convex modules. Examples.- ?4. 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.- ?5. 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.- ?1. 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.- ?2. 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).- ?3. Standard complexes and cohomology groups.- 3.1. Definitions and the basic questions.- 3.2. Some remarks on direct methods.- Notes.- II. Tensor Product.- ?1. Introductory concepts.- 1.1. Universality property. AlgebraiclÈ