##
**Banach algebras and automatic continuity.**
*(English)*
Zbl 0981.46043

London Mathematical Society Monographs. New Series. 24. Oxford: Clarendon Press. xviii, 907 p. (2000).

“This monograph is intended for professional mathematicians and graduate students, especially those working in functional analysis, but also for algebraists. It aims to be a comprehensive account on the interplay between the algebraic and analytic aspects.…The main questions that we address are the following

(1) Let \(A\) and \(B\) be Banach algebras. Under which algebraic conditions on \(A\) and/or \(B\) is it true that every homomorphism from \(A\) into \(B\) is automatically continuous?

(2) Let \(A\) be a Banach algebra, and let \(E\) be a Banach \(A\)-bimodule. Under which algebraic conditions on \(A\) and/or \(E\) is it true that every derivation from \(A\) into \(E\) is automatically continuous?” (from author’s Preface).

While the author lays stress on the automatic continuity problems, the book gives also a fairly full account of the general theory of Banach algebras and can also serve as a (specialized) manual of it. It contains a wealth of many detailed concepts and results in algebra, set theory and foundations of mathematics etc., necessary for further constructions, and so it is self-contained enough. Many results presented in this book are non-available in other books. The proofs are complete, except some results already presented in the “companion volume” published by the author and W. H. Woodin [“An introduction to Independence for Analysts”, Cambridge University Press (1987; Zbl 0629.03030)]. It should be noted that a number of the presented results, especially in Chapter 5, depend upon the continuum hypothesis (CH). The author extends his exposition also onto some topological algebras, especially \((F)\)-algebras, and locally \(m\)-convex algebras. Each section ends with notes including some comments, historical remarks and some related results. There are also formulated or recalled numerous interesting open questions. We quote below some of them.

The book consists of Preface, five chapters, Appendix, References counting 670 positions and Indexes. The first chapter, entitled “Algebraic foundations” gives all necessary information on foundations of mathematics (ordered sets, ordinal and cardinal numbers, continuum hypothesis); semigroups; algebras, ideals and fields; polynomials and formal power series; valuation algebras, derivations; cohomology; involutions. It contains a lot of very specific informations. Chapter 2 is entitled “Banach and topological algebras” and contains following sections: Introduction to normed algebras, Topological algebras, Spectra and Gelfand theory, The functional calculus, Banach algebras of operators, Banach modules, Intertwining maps and derivations, Cohomology, Bounded approximate identities and factorization. Note that in the second section the new concept of a pliable algebra is introduced and studied. Let me mention two problems posed in this chapter:

(1) Let \(D\) be a derivation on a Banach algebra \(A\). Suppose that \(a\in A\) and \(a\cdot D(a)= D(a)\cdot a\). Does it follow that \(D(a)\) is quasinilpotent? The conjecture that the answer is in positive is called the unbounded Kleinecke-Shirokov Conjecture.

(2) Let \(D\) be a derivation on a Banach algebra \(A\). Does it follow that \(D(P)\subset P\) for each primitive ideal \(P\) of \(A\)? The conjecture that the answer is in positive is called the non-commutative Singer-Wermer Conjecture. It is shown later in Chapter 5, that if the answer is negative, then there is a derivative \(D\) on a Banach algebra \(A\) such that \(D(A)\) is not contained in the radical of \(A\) (independent result of M. P. Thomas and Yu. V. Turovskii).

A relatively short Chapter 3 “Banach algebras with an involution” consists of three sections: General theory, \(C^*\)-algebras, and Group algebras. The topological *-algebras are also mentioned here. Chapter 4 “Commutative Banach algebras” gives rather detailed description of many specific commutative Banach algebras taking into account their use in the final Chapter 5 (function algebras, Banach algebras of differentiable and Lipschitz functions, Banach algebras of power series and various convolution algebras). The chapter contains also a classification theory for commutative radical Banach algebras developed by J. R. Esterle, and a basic theory of prime ideals in general commutative Banach algebras. It deals also with the questions of functional continuity in commutative locally \(m\)-convex algebras and of the uniqueness of topology in such algebras.

The climax of the book is in the Chapter 5, Automatic continuity theory (the previous chapters can be treated as a preparation for this one). The first section is devoted to the uniqueness of the norm. It contains the famous result of Barry Johnson stating that every semisimple Banach algebra has a uniquely determined Banach algebra topology (or even the \((F)\)-algebra topology, as shown in the next section). An example shows that the result may fail even if the algebra in question has a one-dimensional radical. Let me quote two problems formulated in this section:

(1) Let \(\theta\) be a homomorphism from a Banach algebra onto a dense subset of a semisimple Banach algebras. Does it follow that \(\theta\) is continuous?

(2) Let \(A\) be a Banach algebra which is a prime algebra, a semiprime algebra, or an integral domain. Does \(A\) necessarily have a unique Banach algebra topology?

The section contains several other (positive or negative) results concerning the uniqueness of topology in Banach algebras.

The next section “The separating space and the stability lemma” is of more technical character and deals also with topological algebras. It contains several results concerning the continuity of derivations and intertwining maps (between the actions of \(A\) on the modules in question) from \(A\) to an \(A\)-module. It contains also further results on uniqueness of the topology for instance a theorem on uniqueness of topology for Banach algebras of power series within the class of \((F)\)-algebras, due to R. J. Loy. It should be mentioned also a result of Johnson and Sinclair stating that every derivation on a semisimple Banach algebra is automatically continuous. The section is concluded with preliminary results on the Singer-Wermer Conjecture stating that the image of a derivation on an arbitrary Banach algebra is contained in its radical. In particular, there is a positive answer to this conjecture in case of a commutative Banach algebra (clue to M. P. Thomas).

The sections 5.3 and 5.4 provide further tools such as the theory of the continuity ideal and the “main boundedness theorem of Bade and Curtis (with their famous result concerning the Wedderburn decomposition of a commutative Banach algebra). A key idea here is that the discontinuity of various discontinuous linear maps is concentrated on a small “discontinuity set” (they are automatically continuous on a large subalgebra). Among the results of these sections we mention the following: Let \(A\) be a strong Ditkin algebra, then each intertwining map from \(A\) into a Banach \(A\)-bimodule is continuous. The same conclusion holds true if \(A\) is a \(C^*\)-algebra, the algebra of all compact operators of a Banach space with the bounded approximation property, or the Volterra algebra. Let \(\Omega\) be a compact space. Then each epimorphism from \(C(\Omega)\) onto a Banach algebra is automatically continuous (Esterle).

There are many more interesting results and problems (for instance the results concerning the continuity of functional calculi) but the following seems to the reviewer to be of a particular interest. Let \(A\) be a commutative Banach algebra which is an integral domain and which does not have a unique complete norm. Then there is a commutative topologically simple Banach algebra (the problem of existence of a commutative Banach algebra without proper closed ideals (or even subalgebras) seems to be one of most challenging questions of the theory of Banach algebras!) Let me note that there is a complete (non-metrizable) commutative locally convex topological algebra without proper closed ideals (Aharon Atzmon), but we do not know any example of a commutative topological algebra without proper closed subalgebras (such an example, due to the reviewer, is known only for semitopological algebras, i.e. algebras with separately continuous multiplication)

. In the next section the author studies the continuity of certain functionals connected with Banach and other topological algebras. These are positive functionals and positive traces on topological *-algebras and higher point derivations on commutative Banach algebras. This includes a result of D. S. Shah about the continuity of positive functionals on \((F)\)-*-algebras, a result of N. Th. Varoupolos stating that each positive functional on a *-Banach algebra with a bounded approximate identity is continuous, and several results on continuity of higher point derivations due to the author and J. P. McClure.

The last but one section “Continuous and discontinuous derivations” deals mostly with cohomologies and the concept of amenability. Among other results it contains the famous result of Barry Johnson stating that a locally compact group \(G\) is amenable if and only if the Banach algebra \(L^1(G)\) is amenable and some results (due mostly to the author) about the existence of discontinuous derivations from Banach algebras of power series into suitable bimodule. Very rich notes to this section give lot of historical informations.

In the last section it is shown that many algebras are normable and that there are discontinuous homomorphism from many (mainly commutative) Banach algebras into other Banach algebras. The first result of this section is the famous result of Graham Allan stating that the algebra of all formal power series can be imbedded into a Banach algebra. As a corollary one can obtain a discontinuous functional calculus for some elements of a radical Banach algebra. It was a surprise to the reviewer to learn that (under assumption of CH) every integral domain of power continuum which is either non-unital or is unital and has a character is normable (Theorem 5.7.19). This result implies (modulo CH) the mentioned earlier result of Allan as well as normability of the Mikusinski convolution algebra of all locally integrable functions on the halfline \({\mathbf R}_+\). The continuum hypothesis if used also for the proof of the famous result obtained independently by the author and J. R. Esterle: there is a discontinuous submultiplicative norm on \(C(\Omega)\) for every infinite compact \(\Omega\) (but there is a model for the set theory, constructed by Woodin, in which an opposite result can be obtained). Several other results in this section also depend upon the continuum hypothesis.

The above report contains only a small fraction of results presented in the book. It deals with actually developed topics in Banach and topological algebras and can be of great use for mathematicians working in this field.

(1) Let \(A\) and \(B\) be Banach algebras. Under which algebraic conditions on \(A\) and/or \(B\) is it true that every homomorphism from \(A\) into \(B\) is automatically continuous?

(2) Let \(A\) be a Banach algebra, and let \(E\) be a Banach \(A\)-bimodule. Under which algebraic conditions on \(A\) and/or \(E\) is it true that every derivation from \(A\) into \(E\) is automatically continuous?” (from author’s Preface).

While the author lays stress on the automatic continuity problems, the book gives also a fairly full account of the general theory of Banach algebras and can also serve as a (specialized) manual of it. It contains a wealth of many detailed concepts and results in algebra, set theory and foundations of mathematics etc., necessary for further constructions, and so it is self-contained enough. Many results presented in this book are non-available in other books. The proofs are complete, except some results already presented in the “companion volume” published by the author and W. H. Woodin [“An introduction to Independence for Analysts”, Cambridge University Press (1987; Zbl 0629.03030)]. It should be noted that a number of the presented results, especially in Chapter 5, depend upon the continuum hypothesis (CH). The author extends his exposition also onto some topological algebras, especially \((F)\)-algebras, and locally \(m\)-convex algebras. Each section ends with notes including some comments, historical remarks and some related results. There are also formulated or recalled numerous interesting open questions. We quote below some of them.

The book consists of Preface, five chapters, Appendix, References counting 670 positions and Indexes. The first chapter, entitled “Algebraic foundations” gives all necessary information on foundations of mathematics (ordered sets, ordinal and cardinal numbers, continuum hypothesis); semigroups; algebras, ideals and fields; polynomials and formal power series; valuation algebras, derivations; cohomology; involutions. It contains a lot of very specific informations. Chapter 2 is entitled “Banach and topological algebras” and contains following sections: Introduction to normed algebras, Topological algebras, Spectra and Gelfand theory, The functional calculus, Banach algebras of operators, Banach modules, Intertwining maps and derivations, Cohomology, Bounded approximate identities and factorization. Note that in the second section the new concept of a pliable algebra is introduced and studied. Let me mention two problems posed in this chapter:

(1) Let \(D\) be a derivation on a Banach algebra \(A\). Suppose that \(a\in A\) and \(a\cdot D(a)= D(a)\cdot a\). Does it follow that \(D(a)\) is quasinilpotent? The conjecture that the answer is in positive is called the unbounded Kleinecke-Shirokov Conjecture.

(2) Let \(D\) be a derivation on a Banach algebra \(A\). Does it follow that \(D(P)\subset P\) for each primitive ideal \(P\) of \(A\)? The conjecture that the answer is in positive is called the non-commutative Singer-Wermer Conjecture. It is shown later in Chapter 5, that if the answer is negative, then there is a derivative \(D\) on a Banach algebra \(A\) such that \(D(A)\) is not contained in the radical of \(A\) (independent result of M. P. Thomas and Yu. V. Turovskii).

A relatively short Chapter 3 “Banach algebras with an involution” consists of three sections: General theory, \(C^*\)-algebras, and Group algebras. The topological *-algebras are also mentioned here. Chapter 4 “Commutative Banach algebras” gives rather detailed description of many specific commutative Banach algebras taking into account their use in the final Chapter 5 (function algebras, Banach algebras of differentiable and Lipschitz functions, Banach algebras of power series and various convolution algebras). The chapter contains also a classification theory for commutative radical Banach algebras developed by J. R. Esterle, and a basic theory of prime ideals in general commutative Banach algebras. It deals also with the questions of functional continuity in commutative locally \(m\)-convex algebras and of the uniqueness of topology in such algebras.

The climax of the book is in the Chapter 5, Automatic continuity theory (the previous chapters can be treated as a preparation for this one). The first section is devoted to the uniqueness of the norm. It contains the famous result of Barry Johnson stating that every semisimple Banach algebra has a uniquely determined Banach algebra topology (or even the \((F)\)-algebra topology, as shown in the next section). An example shows that the result may fail even if the algebra in question has a one-dimensional radical. Let me quote two problems formulated in this section:

(1) Let \(\theta\) be a homomorphism from a Banach algebra onto a dense subset of a semisimple Banach algebras. Does it follow that \(\theta\) is continuous?

(2) Let \(A\) be a Banach algebra which is a prime algebra, a semiprime algebra, or an integral domain. Does \(A\) necessarily have a unique Banach algebra topology?

The section contains several other (positive or negative) results concerning the uniqueness of topology in Banach algebras.

The next section “The separating space and the stability lemma” is of more technical character and deals also with topological algebras. It contains several results concerning the continuity of derivations and intertwining maps (between the actions of \(A\) on the modules in question) from \(A\) to an \(A\)-module. It contains also further results on uniqueness of the topology for instance a theorem on uniqueness of topology for Banach algebras of power series within the class of \((F)\)-algebras, due to R. J. Loy. It should be mentioned also a result of Johnson and Sinclair stating that every derivation on a semisimple Banach algebra is automatically continuous. The section is concluded with preliminary results on the Singer-Wermer Conjecture stating that the image of a derivation on an arbitrary Banach algebra is contained in its radical. In particular, there is a positive answer to this conjecture in case of a commutative Banach algebra (clue to M. P. Thomas).

The sections 5.3 and 5.4 provide further tools such as the theory of the continuity ideal and the “main boundedness theorem of Bade and Curtis (with their famous result concerning the Wedderburn decomposition of a commutative Banach algebra). A key idea here is that the discontinuity of various discontinuous linear maps is concentrated on a small “discontinuity set” (they are automatically continuous on a large subalgebra). Among the results of these sections we mention the following: Let \(A\) be a strong Ditkin algebra, then each intertwining map from \(A\) into a Banach \(A\)-bimodule is continuous. The same conclusion holds true if \(A\) is a \(C^*\)-algebra, the algebra of all compact operators of a Banach space with the bounded approximation property, or the Volterra algebra. Let \(\Omega\) be a compact space. Then each epimorphism from \(C(\Omega)\) onto a Banach algebra is automatically continuous (Esterle).

There are many more interesting results and problems (for instance the results concerning the continuity of functional calculi) but the following seems to the reviewer to be of a particular interest. Let \(A\) be a commutative Banach algebra which is an integral domain and which does not have a unique complete norm. Then there is a commutative topologically simple Banach algebra (the problem of existence of a commutative Banach algebra without proper closed ideals (or even subalgebras) seems to be one of most challenging questions of the theory of Banach algebras!) Let me note that there is a complete (non-metrizable) commutative locally convex topological algebra without proper closed ideals (Aharon Atzmon), but we do not know any example of a commutative topological algebra without proper closed subalgebras (such an example, due to the reviewer, is known only for semitopological algebras, i.e. algebras with separately continuous multiplication)

. In the next section the author studies the continuity of certain functionals connected with Banach and other topological algebras. These are positive functionals and positive traces on topological *-algebras and higher point derivations on commutative Banach algebras. This includes a result of D. S. Shah about the continuity of positive functionals on \((F)\)-*-algebras, a result of N. Th. Varoupolos stating that each positive functional on a *-Banach algebra with a bounded approximate identity is continuous, and several results on continuity of higher point derivations due to the author and J. P. McClure.

The last but one section “Continuous and discontinuous derivations” deals mostly with cohomologies and the concept of amenability. Among other results it contains the famous result of Barry Johnson stating that a locally compact group \(G\) is amenable if and only if the Banach algebra \(L^1(G)\) is amenable and some results (due mostly to the author) about the existence of discontinuous derivations from Banach algebras of power series into suitable bimodule. Very rich notes to this section give lot of historical informations.

In the last section it is shown that many algebras are normable and that there are discontinuous homomorphism from many (mainly commutative) Banach algebras into other Banach algebras. The first result of this section is the famous result of Graham Allan stating that the algebra of all formal power series can be imbedded into a Banach algebra. As a corollary one can obtain a discontinuous functional calculus for some elements of a radical Banach algebra. It was a surprise to the reviewer to learn that (under assumption of CH) every integral domain of power continuum which is either non-unital or is unital and has a character is normable (Theorem 5.7.19). This result implies (modulo CH) the mentioned earlier result of Allan as well as normability of the Mikusinski convolution algebra of all locally integrable functions on the halfline \({\mathbf R}_+\). The continuum hypothesis if used also for the proof of the famous result obtained independently by the author and J. R. Esterle: there is a discontinuous submultiplicative norm on \(C(\Omega)\) for every infinite compact \(\Omega\) (but there is a model for the set theory, constructed by Woodin, in which an opposite result can be obtained). Several other results in this section also depend upon the continuum hypothesis.

The above report contains only a small fraction of results presented in the book. It deals with actually developed topics in Banach and topological algebras and can be of great use for mathematicians working in this field.

Reviewer: Wiesław Tadeusz Zelazko (Warszawa)

### MSC:

46H40 | Automatic continuity |

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

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