# The Homology of Banach and Topological Algebras

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'Et moi *...si j'avait su comment en revenir. One service mathematics has rendered the human race. It has put common sense back je n'y serais point aUe.' it belongs. on the topmost shelf next Jules Verne where to the dusty canister labelled 'discarded non* The series is divergent: therefore we may be sense'. Eric T. Bell able to do something with it. o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

## Product details

• Hardback | 334 pages
• 163.6 x 242.3 x 25.7mm | 703.08g
• Dordrecht, Netherlands
• English
• 1989 ed.
• XX, 334 p.
• 0792302176
• 9780792302179

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(), 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 bimodules.- 3. Derived functors.- 3.1. Definition of derived functors and the long exact sequence.- 3.2. Independence of the choice of resolutions.- 4. 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.- 5. Homological dimensions of modules and algebras.- Notes.- IV. Projectivity.- 1. Some general methods by which projectivity may be checked.- 2. 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.- 3. 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.- 4. 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.- 5. 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.- 1. Koszul resolution.- 1.1. Koszul complex.- 1.2. Koszul resolution for algebras of holomorphic functions.- 2. 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.- 1. The Taylor spectrum and the formulation of the basic theorem.- 2. The complex dominating a module.- 3. Exact complexes of spaces of holomorphic functions. The connection between exactness on the fibres and global exactness.- 4. Construction of the dominating ?ech complex - end of the proof of the basic theorem.- Notes.- VII. Flatness and Amenability.- 1. 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 $$X\mathop{ \otimes }\limits_A^{ \wedge } Y$$. 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.- 2. 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.- 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.