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Elliptic theory and noncommutative geometry. Nonlocal elliptic operators. (English) Zbl 1158.58013
Operator Theory: Advances and Applications 183. Advances in Partial Differential Equations. Basel: Birkhäuser (ISBN 978-3-7643-8774-7/hbk). xii, 224 p. (2008).
The authors are presenting in this book relatively new results, including original ones, generalizing the classical \(K\)-theory and establishing new connections between operator theory and finite geometry. Instead of the old concepts of classical geometry, such as manifolds, metrics or differentiable structures, one operates with their operator-algebraic analogues. The latter are then investigated by using powerful methods from the theory of operator algebras. The operators and associated equations considered in the exposition are of the form \(Du(x)+ D_1u(g(x))= 0\), where \(D\) and \(D_1\) stand for differential operators of nonlocal type on a differentiable manifold. \(g\) stands for a self-mapping of the domain (say \(\Omega\)), where the equation/operators are considered. Only the case when \(g\) is invertible is considered. Equations of the above form are encountered in many mathematical and physical problems. The attention of the authors is focused on the following topics: (1) Elliptic theory on the noncommutative torus and the quantum Hall effect; (2) More general nonlocal operators related to deformations of function algebras on toric manifolds; (3) Nonlocal boundary value problems.
The differential operators involved have coefficients containing shift operators which correspond to the action of a discrete group \(\Gamma\). Under certain technical restrictions, the group \(\Gamma\) can be embedded in an action of a Lie group of diffeomorphisms, which allows to attach to a nonlocal elliptic operator a Fredholm operator in the Hilbert module over the group \(C^*\)-algebra J\(C^*(\Omega)\). This Fredholm operator has an index which is an element of the \(K\)-group of this algebra. The index of the original elliptic operator can be obtained as the image of the index under the mapping induced by the representation \(C^*(\Omega)\to \mathbb{C}\). Various formulas are obtained for the index, a feature that allows the deeper investigation of equations.
Contents: Part I (Nonlocal functions and bundles; Nonlocal elliptic operators; Elliptic operators over \(C^*\)-algebras). Part II (Homotopy classification; Analytic invariants; Bott periodicity; Direct image and index formulas in \(K\)-Theory; Chern character; cohomological index formula; Cohomological formula for the \(\Gamma\)-index; Index of nonlocal operators over \(C^*\)-algebras. Part III (Index formula on the noncommutative torus; An application of higher traces; Index formula for a finite group \(\Gamma\)). Part IV, Appendices (\(C^*\)-algebras; \(K\)-theory of operator algebras: Cyclic homology and cohomology). Three pages of concise bibliographical remarks are closing this highly specialized volume.

58J42 Noncommutative global analysis, noncommutative residues
46L80 \(K\)-theory and operator algebras (including cyclic theory)
58J22 Exotic index theories on manifolds
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
58J05 Elliptic equations on manifolds, general theory
58-02 Research exposition (monographs, survey articles) pertaining to global analysis