The Homology of Banach and Topological Algebras

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. Algebraic tensor product.- 1.2. Tensor products of seminorms.-
2. The tensor product of Banach spaces.- 2.1. Definition and explicit construction.- 2.2. Examples. Tensor multiplication on L1(µ), 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.-
3. 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.-
4. Topological tensor products.- 4.1. The projective and inductive tensor product.- 4.2. Tensor multiplication by an algebra of holomorphic functions and other examples.-
5. 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).-
1. 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.-
2. 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 bimodu