The description of the structure of group C
*-algebras is a difficult problem, but relevant to important new
developments in mathematics, such as non-commutative geometry and quantum groups. Although a significant number of new methods and results have been obtained, until now they have not been available in book form.
This volume provides an introduction to and presents research on the study of group C
*-algebras, suitable for all levels of readers - from graduate students to professional researchers. The introduction provides the essential features of the methods used. In Part I, the author offers an elementary overview - using concrete examples-of using K-homology, BFD functors, and KK-functors to describe group C
*-algebras. In Part II, he uses advanced ideas and methods from representation theory, differential geometry, and KK-theory, to explain two primary tools used to study group C
*-algebras: multidimensional quantization and construction of the index of group C
*-algebras through orbit methods.
T he structure of group C
*-algebras is an important issue both from a theoretical viewpoint and in its applications in physics and mathematics. Armed with the background, tools, and research provided in Methods of Noncommutative Geometry for Group C
*-Algebras, readers can continue this work and make significant contributions to perfecting the theory and solving this problem.
IntroductionThe Scope and an ExampleMultidimensional Orbit MethodsKK-Theory Invariance IndexC*(G)Deformation Quantization and Cyclic TheoriesBibliographical RemarksELEMENTARY THEORY: AN OVERVIEW BASED ON EXAMPLESClassification of MD-GroupsDefinitionsMD CriteriaClassification TheoremBibliographical RemarksThe Structure of C*-Algebras of MD-GroupsThe C*-Algebra of Aff RThe Structure of C*(Aff C)Bibliographical RemarksClassification of MD4-GroupsReal Diamond Group and Semi-Direct Products R x H3Classification TheoremDescription of the Co-Adjoint OrbitsMeasurable MD4-FoliationBibliographical RemarksThe Structure of C*-Algebras of MD4-FoliationsC*-Algebras of Measurable FoliationsThe C*-Algebras of Measurable MD4-FoliationsBibliographic RemarksADVANCED THEORY: MULTIDIMENSIONAL QUANTIZATION AND INDEX OF GROUP C*-ALGEBRASMultidimensional QuantizationInduced Representation. Mackey Method of Small SubgroupsSymplectic Manifolds with Flat Action of Lie GroupsPrequantizationPolarizationBibliographical RemarksPartially Invariant Holomorphly Induced RepresentationsHolomorphly Induced Representations. Lie DerivativeThe Irreducible Representations of Nilpotent Lie GroupsRepresentations of Connected Reductive GroupsRepresentations of Almost Algebraic Lie GroupsThe Trace Formula and the Plancher'el FormulaBibliographical RemarksReduction, Modification, and SuperversionReduction to the Semi-Simple or Reductive CasesMultidimensional Quantization and U(1)-CoveringGlobalization over U(1)-CoveringsQuantization of Mechanical Systems with SupersymmetryBibliographical RemarksIndex of Type I C*-AlgebrasCompact Type Ideals in Type I C*-AlgebrasCanonical Composition seriesIndex of Type I C*-AlgebrasApplication to Lie Group RepresentationsBibliographical RemarksInvariant Index of Group C*-Al
