##
**Banach algebras and the general theory of *-algebras. Volume I: algebras and Banach algebras.**
*(English)*
Zbl 0809.46052

Encyclopedia of Mathematics and Its Applications. 49. Cambridge: Cambridge University Press. xii, 794 p. (1994).

“This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach \(*\)-algebras. This account emphasizes the role of \(*\)- algebra structure and explores the algebraic results which underlie the theory of Banach algebras and \(*\)-algebras. Both volumes contain previously unpublished results.

This first volume is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras.

In both volumes proofs are presented in complete detail at a level accessible to graduate students. In addition, the book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Together these books will become the standard reference for the general theory of \(*\)-algebras.” (cover note).

The present volume consists of a preface, eight chapters, bibliography consisting of 1343 entries, index and symbol index. Each chapter starts with a short introduction. The reader is assumed to have basic knowledge in algebra, topology and functional analysis. The book contains a vast amount of auxiliary material, usually without proofs or with partial proofs only, but enlightened with useful comments. All considered algebras are complex. One of the basic concepts of the book is the concept of a spectral algebra (all Banach algebras are spectral) and many facts known for Banach algebras are formulated and proved here for spectral algebras. Because of very rich historical notes scattered all over the book, it can be of substantial interest to a historian of mathematics working on subjects starting about the end of the last century.

The first chapter “Introduction to normed algebras; Examples” consists of ten sections: Norms and semi-norms on algebras; Double centralizers and extensions; Sums, products and limits; Arens multiplication; Algebras of functions; Matrix algebras; Operator algebras; Group algebras on \(T\); Group algebras; Tensor products. The author introduces here all basic concepts. One of the concepts often used in the book is the concept of a double centralizer algebra of a given algebra (algebra of double multipliers) and it is presented in detail here. A substantial part of the chapter is devoted to examples. It is a vast amount of often advanced material presented in a rather telegraphic style, mostly without proofs. Here the author considers such topics as reflexive and super-reflexive Banach spaces, uniformly convex and uniformly smooth Banach spaces, finite representability, the Radon-Nikodym and super-Radon-Nikodym properties, approximable, compact, weakly compact and strictly singular operators, the approximation property and Enflo’s theorem, nuclear and integral operators, generalized Calkin algebras and Fredholm operators. Further examples concern harmonic analysis, first on the circle \(T\), later on discrete groups and arbitrary locally compact groups. It includes representations of locally compact groups, Fourier and Beurling algebras and semigroup algebras. The Chapter is concluded with tensor products, first in a purely algebraic setting and then with tensor products of Banach algebras. It is illustrated with tensor products of algebras connected with locally compact groups, including a theorem of Varopoulos.

Chapter 2 “The spectrum” consists of 9 sections: Definition of the spectrum, Spectral semi-norms, The Jacobson radical and the Fundamental theorem, Spectral algebras, Spectral subalgebras and Topological divisors of zero, Numerical range in Banach algebras, The spectrum in finite- dimensional algebras, Spectral theory of operators, Topological algebras.

In this chapter the author develops one of the main concepts of the book, namely the concept of a spectral seminorm and a spectral algebra. All Banach algebras are spectral and many results known for Banach algebras are also true for spectral algebras. A spectral seminorm is an algebra seminorm which is greater than or equal to the spectral radius, and a spectral algebra is an algebra having at least one spectral seminorm (usually there may exist many non-equivalent spectral seminorms). The usual properties of spectra and spectral radii are obtained in this more general setting (this includes the analyticity of a resolvent), but, for instance the Vesentini type results about subharmonicity of spectral radius and its logarithm are obtained only for Banach algebras. The Fundamental Theorem (2.3.6) states that if \(\sigma_ 1\) and \(\sigma_ 2\) are two spectral seminorms on \(A\) and if a sequence has limits with respect to both seminorms then their difference lies in the Jacobson radical of \(A\). This result implies immediately the Johnson’s uniqueness of norm theorem. The author poses some open questions on spectral algebras. For instance he asks whether an algebra is spectral if all its maximal commutative subalgebras are spectral (a similar question about Banach algebras was posed by the reviewer). One of the characterizations of spectral seminorms (corollary 2.4.8) says that an algebra seminorm is spectral iff all maximal modular ideals are closed with respect to it. In the following example 2.4.9 (the algebra \(L^ \omega\) of Arens) it should be stated that all its maximal ideals are dense (the word “maximal” is omitted; however Aharon Atzmon constructed a complete locally convex algebra in which all proper ideals are dense – reviewer’s remark). Theorem 2.4.11 states that if an algebra \(A\) has a finite valued spectral radius then this radius is subadditive iff it is submultiplicative iff \(A\) is an almost commutative spectral algebra (commutative modulo its Jacobson radical). Yet another characterization of spectral seminorms is given by theorem 2.5.7. It says that an algebra seminorm is spectral iff every element in the boundary of the set of quasi-invertible elements is a two-sided topological divisor of zero. Theorem 2.5.18 says that an algebra \(A\) is spectral iff \(A\) modulo its Jacobson radical can be imbedded into a Banach algebra (a spectral algebra itself may have no imbedding into a Banach algebra). We shall not describe here the whole content of this chapter.

Chapter 3 “Commutative algebras and functional calculus” consists of 6 sections: Gelfand theory; Shilov boundary, hulls and kernels; Functional calculus; Examples and applications of functional calculus; Multivariable functional calculus; Commutative group algebras.

The Gelfand space, Gelfand transform and the Gelfand radical (kernel of the Gelfand transform) are described here for an arbitrary algebra. Theorem 3.1.5 states that the following are equivalent

(i) \(A\) is an almost commutative spectral algebra,

(ii) \(A\) is a spectral algebra for which the Jacobson and Gelfand radicals coincide,

(iii) the spectral radius is finite valued and subadditive,

(iv) the spectral radius is finite valued and submultiplicative,

(v) every element of \(A\) has a bounded spectrum and \((*)\) \(Sp(a+ b)\subset Sp(a)+ Sp(b)\) for \(a,b\in A\),

(vi) as (v) with \((*)\) replaced by \(Sp(ab)\subset Sp(a)Sp(b)\),

(vii) either the Gelfand space \(\Gamma_ A\) is empty, or it is a locally compact Hausdorff space, and the algebra \(A\) is a separating spectral subalgebra of \(C_ 0(\Gamma_ A)\).

If any of (i)–(vii) holds and \(A\) is unital then \(Sp(a)= a(\Gamma_ A)\). The above gives a number of characterizations of almost commutative algebras among spectral algebras. The Shilov boundary and the hull-kernel topology are considered in the context of spectral algebras and suitable results are the same as in the Banach algebra case. In examples the author presents the constructions of the Stone-Čech and Bohr compactifications. The functional calculus of one variable is treated in the usual way. The author indicates that there may exist several different (discontinuous) functional calculi. The multivariable calculus is based upon the Stokes theorem (the Waelbroeck-Bourbaki approach). While the single variable calculus is given for the Jacobson semisimple spectral algebras (the author does not know whether this assumption can be omitted), the multivariable one is given only for Banach algebras. The author presents such applications as the implicit function theorem, Shilov idempotent, and the Arens-Royden theorem. Another application says that on a semisimple completely regular Banach algebra every algebra norm is spectral. The chapter is closed with the duality theory for locally compact Abelian groups.

Chapter 4 “Ideals, representations and radicals” consists of 8 sections: Ideals and representations, Representations and norms, The Jacobson radical, The Baer radical, The Brown-McCoy or strong radical, Subdirect products, Categorical theory of radicals, History of radicals and examples.

After giving basic algebraic concepts connected with representations of algebras (this includes primitive, prime and quotient ideals) the author describes a less known concept of an ultraprime algebra and ideal. In further discussion on representations of normed or Banach algebras the author often replaces them by spectral algebras. There is also a more advanced discussion on representations on reflexive Banach spaces. The chapter is closed with a thorough discussion of radicals.

A short chapter 5 “Approximate identities and factorization” consists of three sections: Approximate identities and examples, General factorization theorems, Countable factorization theorems. All results are formulated for Banach algebras only.

Chapter 6 “Automatic continuity” consists of 5 sections: Automatic continuity of homomorphisms into \(A\), Automatic continuity of homomorphisms from \(A\), Jordan homomorphisms, Derivations, Jordan derivations.

The chapter contains most known results on the uniqueness of topology problem, continuity of homomorphisms problem and continuity of derivatives problem.

Chapter 7 “Structure spaces” consists of 4 sections: The hull-kernel topology, Completely regular algebras, Primary ideals and Spectral synthesis, Strongly harmonic algebras.

The author studies here the hull-kernel topology (already introduced in chapter 3) for the spaces of all prime, primitive and maximal modular ideals. He also deals with the less popular concept of a strongly harmonic algebra (a weaker concept than strong semisimplicity).

The last chapter 8 “Algebras with minimal ideals” (sections: Finite- dimensional algebras, Minimal ideals and the socle, Algebras of operators with minimal ideals, Modular annihilator algebras, Fredholm theory, Algebras with countable spectrum for elements, Classes of algebras with large socle, Examples) among other things discusses Wedderburn type theorems and counterexamples for Banach algebras, minimal idempotents, and various types of specific algebras such as dual algebras, various types of annihilator algebras and various types of compact algebras.

The tentative contents of volume II consists of the following chapters: 9 “\(*\)-algebras”, 10 “Special \(*\)-algebras”, 11 “Banach \(*\)- algebras”, 12 “Locally compact groups and their \(*\)-algebras”, 13 “Cohomology of Banach algebras”, 14 “K-theory of Banach algebras”. There is no mention of von Neumann algebras.

This first volume is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras.

In both volumes proofs are presented in complete detail at a level accessible to graduate students. In addition, the book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Together these books will become the standard reference for the general theory of \(*\)-algebras.” (cover note).

The present volume consists of a preface, eight chapters, bibliography consisting of 1343 entries, index and symbol index. Each chapter starts with a short introduction. The reader is assumed to have basic knowledge in algebra, topology and functional analysis. The book contains a vast amount of auxiliary material, usually without proofs or with partial proofs only, but enlightened with useful comments. All considered algebras are complex. One of the basic concepts of the book is the concept of a spectral algebra (all Banach algebras are spectral) and many facts known for Banach algebras are formulated and proved here for spectral algebras. Because of very rich historical notes scattered all over the book, it can be of substantial interest to a historian of mathematics working on subjects starting about the end of the last century.

The first chapter “Introduction to normed algebras; Examples” consists of ten sections: Norms and semi-norms on algebras; Double centralizers and extensions; Sums, products and limits; Arens multiplication; Algebras of functions; Matrix algebras; Operator algebras; Group algebras on \(T\); Group algebras; Tensor products. The author introduces here all basic concepts. One of the concepts often used in the book is the concept of a double centralizer algebra of a given algebra (algebra of double multipliers) and it is presented in detail here. A substantial part of the chapter is devoted to examples. It is a vast amount of often advanced material presented in a rather telegraphic style, mostly without proofs. Here the author considers such topics as reflexive and super-reflexive Banach spaces, uniformly convex and uniformly smooth Banach spaces, finite representability, the Radon-Nikodym and super-Radon-Nikodym properties, approximable, compact, weakly compact and strictly singular operators, the approximation property and Enflo’s theorem, nuclear and integral operators, generalized Calkin algebras and Fredholm operators. Further examples concern harmonic analysis, first on the circle \(T\), later on discrete groups and arbitrary locally compact groups. It includes representations of locally compact groups, Fourier and Beurling algebras and semigroup algebras. The Chapter is concluded with tensor products, first in a purely algebraic setting and then with tensor products of Banach algebras. It is illustrated with tensor products of algebras connected with locally compact groups, including a theorem of Varopoulos.

Chapter 2 “The spectrum” consists of 9 sections: Definition of the spectrum, Spectral semi-norms, The Jacobson radical and the Fundamental theorem, Spectral algebras, Spectral subalgebras and Topological divisors of zero, Numerical range in Banach algebras, The spectrum in finite- dimensional algebras, Spectral theory of operators, Topological algebras.

In this chapter the author develops one of the main concepts of the book, namely the concept of a spectral seminorm and a spectral algebra. All Banach algebras are spectral and many results known for Banach algebras are also true for spectral algebras. A spectral seminorm is an algebra seminorm which is greater than or equal to the spectral radius, and a spectral algebra is an algebra having at least one spectral seminorm (usually there may exist many non-equivalent spectral seminorms). The usual properties of spectra and spectral radii are obtained in this more general setting (this includes the analyticity of a resolvent), but, for instance the Vesentini type results about subharmonicity of spectral radius and its logarithm are obtained only for Banach algebras. The Fundamental Theorem (2.3.6) states that if \(\sigma_ 1\) and \(\sigma_ 2\) are two spectral seminorms on \(A\) and if a sequence has limits with respect to both seminorms then their difference lies in the Jacobson radical of \(A\). This result implies immediately the Johnson’s uniqueness of norm theorem. The author poses some open questions on spectral algebras. For instance he asks whether an algebra is spectral if all its maximal commutative subalgebras are spectral (a similar question about Banach algebras was posed by the reviewer). One of the characterizations of spectral seminorms (corollary 2.4.8) says that an algebra seminorm is spectral iff all maximal modular ideals are closed with respect to it. In the following example 2.4.9 (the algebra \(L^ \omega\) of Arens) it should be stated that all its maximal ideals are dense (the word “maximal” is omitted; however Aharon Atzmon constructed a complete locally convex algebra in which all proper ideals are dense – reviewer’s remark). Theorem 2.4.11 states that if an algebra \(A\) has a finite valued spectral radius then this radius is subadditive iff it is submultiplicative iff \(A\) is an almost commutative spectral algebra (commutative modulo its Jacobson radical). Yet another characterization of spectral seminorms is given by theorem 2.5.7. It says that an algebra seminorm is spectral iff every element in the boundary of the set of quasi-invertible elements is a two-sided topological divisor of zero. Theorem 2.5.18 says that an algebra \(A\) is spectral iff \(A\) modulo its Jacobson radical can be imbedded into a Banach algebra (a spectral algebra itself may have no imbedding into a Banach algebra). We shall not describe here the whole content of this chapter.

Chapter 3 “Commutative algebras and functional calculus” consists of 6 sections: Gelfand theory; Shilov boundary, hulls and kernels; Functional calculus; Examples and applications of functional calculus; Multivariable functional calculus; Commutative group algebras.

The Gelfand space, Gelfand transform and the Gelfand radical (kernel of the Gelfand transform) are described here for an arbitrary algebra. Theorem 3.1.5 states that the following are equivalent

(i) \(A\) is an almost commutative spectral algebra,

(ii) \(A\) is a spectral algebra for which the Jacobson and Gelfand radicals coincide,

(iii) the spectral radius is finite valued and subadditive,

(iv) the spectral radius is finite valued and submultiplicative,

(v) every element of \(A\) has a bounded spectrum and \((*)\) \(Sp(a+ b)\subset Sp(a)+ Sp(b)\) for \(a,b\in A\),

(vi) as (v) with \((*)\) replaced by \(Sp(ab)\subset Sp(a)Sp(b)\),

(vii) either the Gelfand space \(\Gamma_ A\) is empty, or it is a locally compact Hausdorff space, and the algebra \(A\) is a separating spectral subalgebra of \(C_ 0(\Gamma_ A)\).

If any of (i)–(vii) holds and \(A\) is unital then \(Sp(a)= a(\Gamma_ A)\). The above gives a number of characterizations of almost commutative algebras among spectral algebras. The Shilov boundary and the hull-kernel topology are considered in the context of spectral algebras and suitable results are the same as in the Banach algebra case. In examples the author presents the constructions of the Stone-Čech and Bohr compactifications. The functional calculus of one variable is treated in the usual way. The author indicates that there may exist several different (discontinuous) functional calculi. The multivariable calculus is based upon the Stokes theorem (the Waelbroeck-Bourbaki approach). While the single variable calculus is given for the Jacobson semisimple spectral algebras (the author does not know whether this assumption can be omitted), the multivariable one is given only for Banach algebras. The author presents such applications as the implicit function theorem, Shilov idempotent, and the Arens-Royden theorem. Another application says that on a semisimple completely regular Banach algebra every algebra norm is spectral. The chapter is closed with the duality theory for locally compact Abelian groups.

Chapter 4 “Ideals, representations and radicals” consists of 8 sections: Ideals and representations, Representations and norms, The Jacobson radical, The Baer radical, The Brown-McCoy or strong radical, Subdirect products, Categorical theory of radicals, History of radicals and examples.

After giving basic algebraic concepts connected with representations of algebras (this includes primitive, prime and quotient ideals) the author describes a less known concept of an ultraprime algebra and ideal. In further discussion on representations of normed or Banach algebras the author often replaces them by spectral algebras. There is also a more advanced discussion on representations on reflexive Banach spaces. The chapter is closed with a thorough discussion of radicals.

A short chapter 5 “Approximate identities and factorization” consists of three sections: Approximate identities and examples, General factorization theorems, Countable factorization theorems. All results are formulated for Banach algebras only.

Chapter 6 “Automatic continuity” consists of 5 sections: Automatic continuity of homomorphisms into \(A\), Automatic continuity of homomorphisms from \(A\), Jordan homomorphisms, Derivations, Jordan derivations.

The chapter contains most known results on the uniqueness of topology problem, continuity of homomorphisms problem and continuity of derivatives problem.

Chapter 7 “Structure spaces” consists of 4 sections: The hull-kernel topology, Completely regular algebras, Primary ideals and Spectral synthesis, Strongly harmonic algebras.

The author studies here the hull-kernel topology (already introduced in chapter 3) for the spaces of all prime, primitive and maximal modular ideals. He also deals with the less popular concept of a strongly harmonic algebra (a weaker concept than strong semisimplicity).

The last chapter 8 “Algebras with minimal ideals” (sections: Finite- dimensional algebras, Minimal ideals and the socle, Algebras of operators with minimal ideals, Modular annihilator algebras, Fredholm theory, Algebras with countable spectrum for elements, Classes of algebras with large socle, Examples) among other things discusses Wedderburn type theorems and counterexamples for Banach algebras, minimal idempotents, and various types of specific algebras such as dual algebras, various types of annihilator algebras and various types of compact algebras.

The tentative contents of volume II consists of the following chapters: 9 “\(*\)-algebras”, 10 “Special \(*\)-algebras”, 11 “Banach \(*\)- algebras”, 12 “Locally compact groups and their \(*\)-algebras”, 13 “Cohomology of Banach algebras”, 14 “K-theory of Banach algebras”. There is no mention of von Neumann algebras.

Reviewer: W.Żelazko (Warszawa)

### MSC:

46Kxx | Topological (rings and) algebras with an involution |

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

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

46Jxx | Commutative Banach algebras and commutative topological algebras |

46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |

46Hxx | Topological algebras, normed rings and algebras, Banach algebras |

46M05 | Tensor products in functional analysis |

46L05 | General theory of \(C^*\)-algebras |

47L30 | Abstract operator algebras on Hilbert spaces |

47L10 | Algebras of operators on Banach spaces and other topological linear spaces |