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  • 1
    Online Resource
    Online Resource
    An Introduction to C*-Algebras and Noncommutative Geometry (2024)
    Cham :Springer International Publishing :
    Details
    UID:
    almahu_9949851978002882
    Format: XIII, 539 p. 27 illus. , online resource.
    Edition: 1st ed. 2024.
    ISBN: 9783031598500
    Series Statement: Birkhäuser Advanced Texts Basler Lehrbücher,
    Content: This is the first textbook on C*-algebra theory with a view toward Noncommutative Geometry. Moreover, it fills a gap in the literature, providing a clear and accessible account of the geometric picture of K-theory and its relation to the C*-algebraic picture. The text can be used as the basis for a graduate level or a capstone course with the goal being to bring a relative novice up to speed on the basic ideas while offering a glimpse at some of the more advanced topics of the subject. Coverage includes C*-algebra theory, K-theory, K-homology, Index theory and Connes' Noncommuntative Riemannian geometry. Aimed at graduate level students, the book is also a valuable resource for mathematicians who wish to deepen their understanding of noncommutative geometry and algebraic K-theory. A wide range of important examples are introduced at the beginning of the book. There are lots of excellent exercises and any student working through these will benefit significantly. Prerequisites include a basic knowledge of algebra, analysis, and a bit of functional analysis. As the book progresses, a little more mathematical maturity is required as the text discusses smooth manifolds, some differential geometry and elliptic operator theory, and K-theory. The text is largely self-contained though occasionally the reader may opt to consult more specialized material to further deepen their understanding of certain details.
    Note: An introduction to C*-algebras -- An Introduction to Index Theory and Noncommutative Geometry -- Spectral Theory and Representation -- Positivity, Representations, Tensor Products and Ideals in C*-algebras -- Module theory of C*-algebras -- Morita Equivalence -- Topological K-theory and Clifford Algebras -- K-theory for C*algebras -- The Index Theorem of Atiyah and Singer -- K-homology and Noncommutative Geometry -- An Introduction to KK-theory -- Bibliography.
    In: Springer Nature eBook
    Additional Edition: Printed edition: ISBN 9783031598494
    Additional Edition: Printed edition: ISBN 9783031598517
    Additional Edition: Printed edition: ISBN 9783031598524
    Language: English
    DOI: 10.1007/978-3-031-59850-0
    URL: https://doi.org/10.1007/978-3-031-59850-0
    URL: Volltext  (URL des Erstveröffentlichers)
    URL: lizenzpflichtig
    URL: Cover
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  • 2
    Online Resource
    Online Resource
    An Introduction to C*-Algebras and Noncommutative Geometry. (2024)
    Cham :Springer International Publishing AG,
    Details
    UID:
    edoccha_9961612699502883
    Format: 1 online resource (548 pages)
    Edition: 1st ed.
    ISBN: 9783031598500
    Series Statement: Birkhäuser Advanced Texts Basler Lehrbücher Series
    Note: Intro -- Preface -- Contents -- 1 An Introduction to C*-Algebras -- 1.1 The Definition of C*-Algebra -- 1.2 The C*-Algebra of Bounded Operators on a Hilbert Space -- 1.3 Group C*-Algebras -- 1.4 C*-Algebras of the Integers and the Circle -- 1.5 C*-Algebras of Finite Groups -- 1.6 The Compact Operators -- 1.7 Inductive Limits of C*-Algebras -- 1.8 Crossed Product C*-Algebras -- 1.9 The C*-Algebra and Fourier Transform for the Group of Real Numbers -- 2 An Introduction to Index Theory and Noncommutative Geometry -- 2.1 Trace-Class and Schatten Classes of Compact Operators -- 2.2 The Toeplitz Algebra -- 2.3 The Toeplitz Trace Theorem -- 2.4 The Calkin Algebra -- 2.5 General Properties of the Fredholm Index -- 2.6 The Toeplitz Index Theorem -- 3 Spectral Theory and Representations -- 3.1 Spectrum in a Banach Algebra -- 3.2 The Holomorphic Functional Calculus -- 3.3 Characters and Gelfand's Theorem -- 3.4 Gelfand's Theorem -- 3.5 Functional Calculus, Isospectral Subalgebras -- 4 Positivity, Representations, Tensor Products, and Ideals in C*-Algebras -- 4.1 Positivity and States -- 4.2 The GNS Theorem -- 4.3 Generalities About Completions of *-Algebras -- 4.4 Ideals and Quotients of C*-Algebras -- 4.5 Tensor Products of C*-Algebras -- 4.6 Structure of Crossed Products by Proper Actions of Discrete Groups -- 5 Module Theory of C*-Algebras -- 5.1 Vector Bundles -- 5.2 Finitely Generated Projective (f.g.p.) Modules and Vector Bundles: Swan's Theorem -- 5.3 Multiplier Algebras -- 5.4 Hilbert Modules -- 5.5 Operators on Hilbert Modules, Tensor Products, and Applications -- 6 Morita Equivalence -- 6.1 Morita Equivalence -- 6.2 Finitely Generated Projective Modules and Morita Correspondences -- 6.3 Morita Correspondences from Equivariant Vector Bundles -- 6.4 Morita Correspondences and Representations of Finite Groups. , 6.5 Remarks on Compact Noncommutative Spaces -- 6.6 Morita Correspondences Between Irrational Tori -- 6.7 Morita Equivalence and Asymptotics in Hyperbolic Geometry -- 7 Topological K-Theory and Clifford Algebras -- 7.1 The Definition of K-Theory, the K-Theory of the Circle -- 7.2 Vector Bundles on Smooth Manifolds -- 7.3 Functoriality and Homotopy-Invariance -- 7.4 K-Theory for Noncompact Spaces, Higher K-groups -- 7.5 The Long Exact Sequence of a Pair -- 7.6 Bott Periodicity, the 6-Term Exact Sequence -- 7.7 Spin Geometry and Clifford Algebras -- 7.8 Representation Theory of Clifford Algebras -- 7.9 Clifford Algebras and K-Theory: The Bott-Shapiro Theorem -- 7.10 K-Orientations and the Thom Isomorphism Theorem -- 8 K-Theory for C*-Algebras -- 8.1 Basic Definitions of C*-Algebra K-Theory -- 8.2 Morita Invariance and Applications -- 8.3 Higher K-Theory, Loops and Unitaries -- 8.4 The Long Exact Sequence -- 8.5 The Long Exact Sequence -- 8.6 Examples of the Connecting Homomorphism -- 8.7 The External Product Operation on K-Theory -- 8.8 The Bott Periodicity Theorem -- 8.9 Some Orbifold K-Theory Computations -- 9 The Index Theorem of Atiyah and Singer -- 9.1 Differential Operators on Euclidean Space -- 9.2 Differential Operators on Manifolds -- 9.3 Analytic Aspects of Elliptic Operators -- 9.4 Dirac Operators -- 9.5 Bounded Transforms of Dirac Operators -- 9.6 The K-Theory Index Theorem(s) -- 10 K-Homology and Noncommutative Geometry -- 10.1 Fredholm Modules and Their Pairing with K-Theory -- 10.2 Cyclic Cohomology -- 10.3 Finitely Summable Fredholm Modules and Connes Character Formula -- 10.4 Fredholm Modules from Boundary Actions of Hyperbolic Groups -- 10.5 Fredholm Modules from Extensions -- 10.6 Spectral Cycles and Fredholm Modules -- 10.7 The Heat Equation Proof of the Atiyah-Singer Index Theorem. , 10.8 The Atiyah-Singer and Connes-Moscovici Local Index Theorems -- 10.9 Zeta Functions and the Local Index Theorem for the Circle -- 10.10 Heisenberg Spectral Cycles and Irrational Rotation -- 10.11 The Harmonic Oscillator Residue Trace -- 10.12 The Local Index Formula for the Heisenberg Cycles -- 11 An Introduction to KK-Theory -- 11.1 Basic Definitions of KK -- 11.2 Category Structure of KK -- 11.3 The Axiomatic Approach to the Kasparov Product -- 11.4 The Bott Periodicity Theorem in KK-Theory -- 11.5 Equivariant Bott Periodicity and the K-Theory of Crossed Products -- Bibliography.
    Additional Edition: Print version: Emerson, Heath An Introduction to C*-Algebras and Noncommutative Geometry Cham : Springer International Publishing AG,c2024 ISBN 9783031598494
    Language: English
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