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Several complex variables and Banach algebras. 3rd ed. (English) Zbl 0894.46037

Graduate Texts in Mathematics. 35. New York, NY: Springer. xii, 253 p. (1998).
Previous editions have been reviewed both in the Mathematical Reviews [MR 46#672, MR 52#15021] and in the Zentralblatt [Zbl 0281.46052, Zbl 0336.46055; the last one is the most detailed]. The changes (quite substantial) against the second edition are as follows. The basic material (Chapters 1-10) has been left intact, and so have the sections dealing with Hörmander’s \(\bar\partial\)-theory (Chapters 16-17), submanifolds of high dimension (Ch. 18), and the Arens-Royden theorem (Ch. 15). The remaining chapters 11-14 and 19-21, dealing in turn with analytic discs in the maximal ideal space, algebras of analytic functions, approximation on curves, approximation on discs, generators of certain function algebras on the ball, fibers over a plane domain, and examples of hulls, have been replaced by new material. The new Chapter 11 is devoted to maximum modulus algebras, Chapter 12 to polynomial hulls of curves and arcs, Chapter 13 to integral kernels, and Chapter 14 to “perturbed” Stone-Weierstrass theorems. Chapter 19 describes the work of Harvey and Lawson on constructing a complex manifold with a prescribed boundary; Chapter 20 deals with polynomial hulls of sets over the circle, and Chapters 21 (Areas), 22 (Topology of hulls) and 23 (Pseudoconvex sets in \(\mathbf C^n\)) with some geometric, topological, and pseudoconvexity properties of polynomial hulls, respectively. Chapter 24 (Examples) combines some examples from the omitted chapters 13, 20 and 21 of the second edition, and also contains several new ones. Some historical comments and notes on recent developments are collected in Chapter 25, and a number of classical results and other prerequisites are gathered in an appendix (Chapter 26). Compared to the second edition, the authors have added a lot of new bibliography, and have substantially improved the index.
Needless to say, the book has retained all the pedagogical excellence of the first two editions of this classic.
Reviewer: M.Engliš (Praha)

MSC:

46Jxx Commutative Banach algebras and commutative topological algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
32Exx Holomorphic convexity
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32Dxx Analytic continuation
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