On induced representations of discrete groups

Preface Chapter 3. Cyclic Cohomology and Differential Geometry 1. Cyclic Cohomology 2. Examples 3. Pairing of Cyclic Cohomology with K-Theory 4. The Higher Index Theorem for Covering Spaces 5. The Novikov Conjecture for Hyperbolic Groups 6. Factors of Type III, Cyclic Cohomology and the Godbillon-Vey Invariant 7. The Transverse Fundamental Class for Foliations and Geometric Corollaries Appendix A. The Cyclic Category Λ Appendix B. Locally Convex Algebras Appendix C. Stability under Holomorphic Functional Calculus Chapter 4. Quantized Calculus 1. Quantized Differential Calculus and Cyclic Cohomology 2. The Dixmier Trace and the Hochschild Class of the Character 3. Quantized Calculus in One Variable and Fractal Sets 4. Conformal Manifolds 5. Fredholm Modules and Rank-One Discrete Groups 6. Elliptic Theory on the Noncommutative Torus T 2 θ and the Quantum Hall Effect 7. Entire Cyclic Cohomology 8. The Chern Character of θ-summable Fredholm Modules 9. θ-summable K-cycles, Discrete Groups and Quantum Field Theory Appendix A. Kasparov's Bivariant Theory Appendix B. Real and Complex Interpolation of Banach Spaces Appendix C. Normed Ideals of Compact Operators Appendix D. The Chern Character of Deformations of Algebras Chapter 5. Operator algebras 1. The Papers of Murray and von Neumann 2. Representations of C * -algebras 3. The Algebraic Framework for Noncommutative Integration and the Theory of Weights 4. The Factors of Powers, Araki and Woods, and of Krieger 5. The Radon-Nikodým Theorem and Factors of Type III λ 6. Noncommutative Ergodic Theory 7. Amenable von Neumann Algebras 8. The Flow of Weights: Mod(M )This book is the English version of the French "Géométrie non commutative" published by InterEditions Paris (1990). After the initial translation by S.K. Berberian, a considerable amount of rewriting was done and many additions made, multiplying by 3.8 the size of the original manuscript. In particular the present text contains several unpublished results. My thanks go first of all to Cécile whose patience and care for the manuscript have been essential to its completion. This second version of the book greatly benefited from the important modifications suggested by M.

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“…In this section we assume dim(V σ ) = 1, although some of the results still will be true in greater generality, in particular the results about the set B, cf. [2,3].…”

Section: Hecke Pairs and Schlichting Completionsmentioning

confidence: 99%

“…Proof. Clearly ε x ∈ H σ (G, H) and is well-defined (these functions are essentially the elementary intertwining operators from [2]). Since ε h 0 xk 0 (hxk) = σ(h 0 )σ(k 0 )ε x (hxk) for all h 0 , k 0 ∈ H, different choices of representatives for the double coset HxH give rise to functions ε h 0 xk 0 which are scalar multiples of ε x , and since distinct double cosets do not support a common ε x the lemma follows.…”

Section: Hecke Pairs and Schlichting Completionsmentioning

confidence: 99%

Generalized Hecke Algebras and C*-Completions

Landstad

Larsen

2009

Int. J. Math.

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For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range we consider a generalised Hecke algebra Hσ(G, H), which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection pσ ∈ Cc(Gσ, B(Vσ)) such that Hσ(G, H) is isomorphic to pσCc(Gσ, B(Vσ))pσ. We study the structure and properties of C * -completions of the generalised Hecke algebra arising from this corner realisation, and via Morita-Fell-Rieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of C * (Gσ, B(Vσ)). By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + bgroups and the Heisenberg group.

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On induced representations of discrete groups

Cited by 9 publications

References 15 publications

On discrete groups of Euclidean isometries: representation theory, harmonic analysis and splitting properties

On discrete groups of Euclidean isometries: representation theory, harmonic analysis and splitting properties

Noncommutative Geometry

Generalized Hecke Algebras and C*-Completions

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