Homology in Banach and topological algebras. (Gomologiya v banakhovykh i topologicheskikh algebrakh).

*(Russian)*Zbl 0608.46046
Moskva: Izdatel’stvo Moskovskogo Universiteta. 288 p. R. 3.50 (1986).

In the last two decades the use of homological methods in several branches of functional, complex of harmonic analysis became a necessity. For instance, this tendency may be illustrated by the following directions: the Hochschild (co)homology of Banach and topological algebras [cf. B. E. Johnson, ”Cohomology in Banach algebras”, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014) and A. Connes, ”Non-commutative differential geometry”, Publ. Math. Inst. Haut. Etud. Sci. 62, 257-360 (1986; Zbl 0592.46056)], base change and Künneth type formulae in analytic geometry [cf. Séminaire Douady-Verdier, Astérisque, No.16 (1974)], the multidimensional spectral theory [cf. J. L. Taylor, Adv. Math. 9, 183-252 (1972; Zbl 0271.46041)] or the general theory of topological vector spaces [cf. V. P. Palamodov, Usp. Mat. Nauk 26, No.1(157), 3-65 (1971; Zbl 0247.46070)] and so on. The book under review is both an introduction to these topics and a comprehensive treatise on projective and flat modules oer topological (mainly Banach) algebras. The book is divided into two parts.

Part one is preparatory and begins with Chapter 0 where facts from linear algebra, functional analysis and homological algebra are carefully recalled and isolated (without proofs). Chapter 1 deals with the interpretation of the lower Hochschild cohomology groups of a Banach algebra with coefficients in a Banach bimodule. This approach repeats the historical development of the field until 1980 and provides a stimulating motivation for the next chapters. Chapter 2 is an excursion through the theory of topological tensor products and related subjects, as for instance the approximation property of Banach spaces.

Part two, basic one, begins with Chapter 3 where the framework is established and the main objects to be investigated are introduced. Roughly, one works in the category of Fréchet A-modules over a Fréchet algebra A, with admissible (short) exact sequences of A-modules as exact sequences which are in addition topologically \({\mathbb{C}}\)-split. It turns out that under this assumption there are sufficiently many projective (in the Banach space case also injective) modules. This is an example, non-trivial by its consequences, of a relative homology theory as introduced by S. Eilenberg and J. C. Moore in ”Foundations of relative homological algebra”, Mem. Am. Math. Soc. 55, 39 p. (1965; Zbl 0129.01101). The topological bivariant functors \(Hom_ A(M,N)\), M\({\hat \otimes}_ AN\) are then introduced together with their derived functors Ext\(^._ A(M,N)\) and Tor\(^ A_.(M,N)\). Chapter 4 deals exclusively and comprehensively with questions related to projectiveness. Several sufficient (and necessary) conditions for an A-module to be projective are proved. Many interesting results which are due to the author and his school are presented for the first time into a volume. Among these we reproduce the following statement: let \(\Omega\) be a locally compact space and let \(I\subset C_ 0(\Omega)\) be a closed ideal in the commutative Banach algebra of continuous functions on \(\Omega\), vanishing at \(\infty\). The ideal I is projective if and only if its maximal spectrum is paracompact. Other theorems refer to group algebras or operator algebras. Chapter 5 is devoted to the computation of various homological dimensions of algebras and modules over them. Here things seem to be at a quite incipient stage, a list with open problems ending this chapter. Chapter 6 presents with minor modifications J. L. Taylor’s method for deriving the existence of an analytic functional calculus for commutative systems of operators from the theory of topological Tors. Behind this lies a natural localization idea which could be helpful in many other contexts. Chapter 7 is the last of the book and is concerned with flatness and amenability. Criteria for an A-module to be topologically A-flat are proved, and then they are applied to group algebras or uniform algebras, where a simple relationship between flatness and amenability holds. However, the recent results on amenable \(C^*\)- and von Neumann algebras are not included in this chapter. The book ends by two short complements on paracompact spaces and respectively on invariant means on groups.

The author, one of the creators of the topological relative homology, addresses in this book both to beginners as well as to experts. Written on the basis of the special course delivered at Moscow University, the book is an excellent introductory course in topological homology. On the other hand, the analysts find in this text a rich material concerning the homological aspects of their actual field of interest, completed by accurate comments. Far from being exhaustive, the book certainly fills a gap in the literature of the domain.

Part one is preparatory and begins with Chapter 0 where facts from linear algebra, functional analysis and homological algebra are carefully recalled and isolated (without proofs). Chapter 1 deals with the interpretation of the lower Hochschild cohomology groups of a Banach algebra with coefficients in a Banach bimodule. This approach repeats the historical development of the field until 1980 and provides a stimulating motivation for the next chapters. Chapter 2 is an excursion through the theory of topological tensor products and related subjects, as for instance the approximation property of Banach spaces.

Part two, basic one, begins with Chapter 3 where the framework is established and the main objects to be investigated are introduced. Roughly, one works in the category of Fréchet A-modules over a Fréchet algebra A, with admissible (short) exact sequences of A-modules as exact sequences which are in addition topologically \({\mathbb{C}}\)-split. It turns out that under this assumption there are sufficiently many projective (in the Banach space case also injective) modules. This is an example, non-trivial by its consequences, of a relative homology theory as introduced by S. Eilenberg and J. C. Moore in ”Foundations of relative homological algebra”, Mem. Am. Math. Soc. 55, 39 p. (1965; Zbl 0129.01101). The topological bivariant functors \(Hom_ A(M,N)\), M\({\hat \otimes}_ AN\) are then introduced together with their derived functors Ext\(^._ A(M,N)\) and Tor\(^ A_.(M,N)\). Chapter 4 deals exclusively and comprehensively with questions related to projectiveness. Several sufficient (and necessary) conditions for an A-module to be projective are proved. Many interesting results which are due to the author and his school are presented for the first time into a volume. Among these we reproduce the following statement: let \(\Omega\) be a locally compact space and let \(I\subset C_ 0(\Omega)\) be a closed ideal in the commutative Banach algebra of continuous functions on \(\Omega\), vanishing at \(\infty\). The ideal I is projective if and only if its maximal spectrum is paracompact. Other theorems refer to group algebras or operator algebras. Chapter 5 is devoted to the computation of various homological dimensions of algebras and modules over them. Here things seem to be at a quite incipient stage, a list with open problems ending this chapter. Chapter 6 presents with minor modifications J. L. Taylor’s method for deriving the existence of an analytic functional calculus for commutative systems of operators from the theory of topological Tors. Behind this lies a natural localization idea which could be helpful in many other contexts. Chapter 7 is the last of the book and is concerned with flatness and amenability. Criteria for an A-module to be topologically A-flat are proved, and then they are applied to group algebras or uniform algebras, where a simple relationship between flatness and amenability holds. However, the recent results on amenable \(C^*\)- and von Neumann algebras are not included in this chapter. The book ends by two short complements on paracompact spaces and respectively on invariant means on groups.

The author, one of the creators of the topological relative homology, addresses in this book both to beginners as well as to experts. Written on the basis of the special course delivered at Moscow University, the book is an excellent introductory course in topological homology. On the other hand, the analysts find in this text a rich material concerning the homological aspects of their actual field of interest, completed by accurate comments. Far from being exhaustive, the book certainly fills a gap in the literature of the domain.

Reviewer: M.Putinar

##### MSC:

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46M10 | Projective and injective objects in functional analysis |

46H10 | Ideals and subalgebras |

46H30 | Functional calculus in topological algebras |

46M15 | Categories, functors in functional analysis |

46M05 | Tensor products in functional analysis |