Toeplitz operators and index theory in several complex variables.

*(English)*Zbl 0957.47023
Operator Theory: Advances and Applications. 81. Basel: Birkhäuser. 481 p. (1996).

This book gives a comprehensive presentation of modern results on Toeplitz operator algebras in the setting of several complex variables. The book is self-contained, i.e., it provides all necessary preliminary material, and treats almost all known results (to the moment of its writing) connected with the theme. The passage from the classical results in one complex variable to the multivariable setting is nontrivial. Both the results and tools used depend much on the type of underlying domain of holomorphy. The main qualitative difference (in several variables) is that the quotient algebra of the Toeplitz operator \(C^*\)-algebra modulo the ideal of compact operators may have (and as a rule has) infinite-dimensional irreducible representations, and the structure of these representations depends on and reflects geometrical properties of the boundary of the underlying domain.

The first, introductory, chapter, “Multi-variable complex analysis and domains of holomorphy”, is devoted to a detailed description of various types of domains of holomorphy in several complex variables – the underlying stage for the Toeplitz operators studied in this book. The author starts with strictly pseudoconvex domains, the most direct analogue of domains in one-dimensional complex analysis. The multidimensional analogues of a half-plane – the tubular domains – are considered over homogeneous symmetric (convex) cones. This permits one to characterize them algebraically in terms of Jordan algebras. The (type-II) Siegel domains are considered here as well. The polycircular (or Reinhardt) domains admit an action of the \(r\)-torus. This action leads to a very important foliation describing the boundary geometry and characterizing later on the operator-theoretic behavior of Toeplitz operators. The basic geometric feature of symmetric domains is a stratification of their boundary. To characterize algebraically these domains as well as their “boundary components” the author introduces the so-called Jordan triples. The simultaneous generalization of polycircular and symmetric domains – the so-called \(K\)-circular domains and their special subclass of \(S\)-bicircular domains – is considered as well.

Chapter 2, “Harmonic analysis on Hilbert spaces of holomorphic functions”, describes the spaces on which the Toeplitz operators act, the Hardy and the Bergman spaces of analytic functions associated with the domains of holomorphy introduced above, as well as the corresponding Szegő and Bergman projections. The different nature of the domains requires different techniques and tools. The most classical tools – partial differential methods (\(\overline\partial\)-problem, Kohn-Laplacian) – are used in the case of strictly pseudoconvex domains. In several variables the domains in general do not have a smooth boundary and thus one cannot continue to use partial differential methods. But these domains have instead sufficiently rich groups of automorphisms, and thus the description of spaces of analytic functions (the Hardy and the Bergman spaces) are based on different versions of Fourier analysis. The Szegő and Bergman projections are characterized in terms of their kernels; for symmetric domains this is done using the underlying Jordan algebraic structure.

Chapter 3, “Multiplier \(C^*\)-algebras and their representations”, is devoted to the study of some auxiliary \(C^*\)-algebras (of multipliers) containing the projection onto the corresponding space of analytic functions. The domains under consideration admit groups of automorphisms and the multiplier algebra (or algebras of convolution operators under Fourier transform) are certain completions of the corresponding group \(C^*\)-algebras. The reason for introducing these algebras is to study the (Szegő and Bergman) projections in an algebraic setting and to understand the connection between a stratification of the boundary of the underlying domain and a geometric realization of the spectrum of the corresponding \(C^*\)-algebra.

Chapter 4, “Toeplitz operators and Toeplitz \(C^*\)-algebras”, is central to the book and is devoted to the structural theory of \(C^*\)-algebras generated by Toeplitz operators with continuous symbols, in both the Hardy and the Bergman space settings. For different types of underlying domains, the systems of irreducible representations of the corresponding Toeplitz \(C^*\)-algebras as well as the methods of study are quite different and reflect essentially the geometric properties of the domains. In the case of strictly pseudoconvex domains the Toeplitz operators are essentially normal, the \(C^*\)-algebra generated by them is irreducible, and the quotient algebra with respect to the ideal of compact operators is commutative and thus has one-dimensional irreducible representations only. The main qualitative result for more general domains (tubular, polycircular, bounded symmetric and Siegel) is that the corresponding Toeplitz \(C^*\)-algebras are solvable of length \(r\), where \(r\) is the geometric rank of the underlying domain; the irreducible representations (generally speaking infinite-dimensional) are parametrized by the points of holomorphic faces of the domain. The approach based on groupoid-theoretic methods is used in the case of tubular and polycircular domains, while for bounded symmetric domains the author uses the theory of Hopf \(C^*\)-algebras and (co)crossed products.

Finally, Chapter 5, “Index theory for multivariable Toeplitz operator”, treats the topological aspects of Toeplitz \(C^*\)-algebras in \(K\)-theoretical terms. For algebras solvable of length 1 (the case of strictly pseudoconvex domains and domains of finite type) the Fredholm index formulas are obtained. For nonsmooth domains with stratified boundary geometry one has instead a family of “\(K\)-theory valued” Fredholm indices, parametrized by holomorphic faces of the underlying domain.

The first, introductory, chapter, “Multi-variable complex analysis and domains of holomorphy”, is devoted to a detailed description of various types of domains of holomorphy in several complex variables – the underlying stage for the Toeplitz operators studied in this book. The author starts with strictly pseudoconvex domains, the most direct analogue of domains in one-dimensional complex analysis. The multidimensional analogues of a half-plane – the tubular domains – are considered over homogeneous symmetric (convex) cones. This permits one to characterize them algebraically in terms of Jordan algebras. The (type-II) Siegel domains are considered here as well. The polycircular (or Reinhardt) domains admit an action of the \(r\)-torus. This action leads to a very important foliation describing the boundary geometry and characterizing later on the operator-theoretic behavior of Toeplitz operators. The basic geometric feature of symmetric domains is a stratification of their boundary. To characterize algebraically these domains as well as their “boundary components” the author introduces the so-called Jordan triples. The simultaneous generalization of polycircular and symmetric domains – the so-called \(K\)-circular domains and their special subclass of \(S\)-bicircular domains – is considered as well.

Chapter 2, “Harmonic analysis on Hilbert spaces of holomorphic functions”, describes the spaces on which the Toeplitz operators act, the Hardy and the Bergman spaces of analytic functions associated with the domains of holomorphy introduced above, as well as the corresponding Szegő and Bergman projections. The different nature of the domains requires different techniques and tools. The most classical tools – partial differential methods (\(\overline\partial\)-problem, Kohn-Laplacian) – are used in the case of strictly pseudoconvex domains. In several variables the domains in general do not have a smooth boundary and thus one cannot continue to use partial differential methods. But these domains have instead sufficiently rich groups of automorphisms, and thus the description of spaces of analytic functions (the Hardy and the Bergman spaces) are based on different versions of Fourier analysis. The Szegő and Bergman projections are characterized in terms of their kernels; for symmetric domains this is done using the underlying Jordan algebraic structure.

Chapter 3, “Multiplier \(C^*\)-algebras and their representations”, is devoted to the study of some auxiliary \(C^*\)-algebras (of multipliers) containing the projection onto the corresponding space of analytic functions. The domains under consideration admit groups of automorphisms and the multiplier algebra (or algebras of convolution operators under Fourier transform) are certain completions of the corresponding group \(C^*\)-algebras. The reason for introducing these algebras is to study the (Szegő and Bergman) projections in an algebraic setting and to understand the connection between a stratification of the boundary of the underlying domain and a geometric realization of the spectrum of the corresponding \(C^*\)-algebra.

Chapter 4, “Toeplitz operators and Toeplitz \(C^*\)-algebras”, is central to the book and is devoted to the structural theory of \(C^*\)-algebras generated by Toeplitz operators with continuous symbols, in both the Hardy and the Bergman space settings. For different types of underlying domains, the systems of irreducible representations of the corresponding Toeplitz \(C^*\)-algebras as well as the methods of study are quite different and reflect essentially the geometric properties of the domains. In the case of strictly pseudoconvex domains the Toeplitz operators are essentially normal, the \(C^*\)-algebra generated by them is irreducible, and the quotient algebra with respect to the ideal of compact operators is commutative and thus has one-dimensional irreducible representations only. The main qualitative result for more general domains (tubular, polycircular, bounded symmetric and Siegel) is that the corresponding Toeplitz \(C^*\)-algebras are solvable of length \(r\), where \(r\) is the geometric rank of the underlying domain; the irreducible representations (generally speaking infinite-dimensional) are parametrized by the points of holomorphic faces of the domain. The approach based on groupoid-theoretic methods is used in the case of tubular and polycircular domains, while for bounded symmetric domains the author uses the theory of Hopf \(C^*\)-algebras and (co)crossed products.

Finally, Chapter 5, “Index theory for multivariable Toeplitz operator”, treats the topological aspects of Toeplitz \(C^*\)-algebras in \(K\)-theoretical terms. For algebras solvable of length 1 (the case of strictly pseudoconvex domains and domains of finite type) the Fredholm index formulas are obtained. For nonsmooth domains with stratified boundary geometry one has instead a family of “\(K\)-theory valued” Fredholm indices, parametrized by holomorphic faces of the underlying domain.

Reviewer: N.L.Vasilevskii (MR 97f:47022)

##### MSC:

47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |

47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |

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

32T05 | Domains of holomorphy |

32T15 | Strongly pseudoconvex domains |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |

46E20 | Hilbert spaces of continuous, differentiable or analytic functions |