1 - Group C*-Algebras, C*-Correspondences and K-Theory
Published online by Cambridge University Press: 06 November 2024
Summary
We start by introducing C*-algebras associated with locally compact groups. Next, the theory of Hilbert modules, C*-correspondences, crossed-product algebras, and Morita equivalence are discussed. This is followed with applications to Mackey’s theory of induction of representations and Howe’s theory of theta correspondence. Next, K-theory and (equivariant) KK-theory are introduced. Connections between isolated points in the unitary dual and generators of the K-theory of C*-algebras of liminal groups are discussed.
Keywords
- Type
- Chapter
- Information
- K-Theory and Representation Theory , pp. 1 - 36Publisher: Cambridge University PressPrint publication year: 2024
References
Baaj, S. and Julg, P.. Théorie bivariante de Kasparov et opérateurs non bornés dans les C∗-modules hilbertiens. C. R. Acad. Sci. Paris Sér. I Math. 296 (21) (1983), 875–878.Google Scholar
Baum, P., Connes, A., and Higson, N.. Classifying space for proper actions and K-theory of group C∗-algebras. In C∗-algebras: 1943–1993 a fifty year celebration (ed. Doran, R., San Antonio, TX, 1993), 241–291, Contemporary Mathematics, 167, American Mathematical Society, Providence, RI, 1994.Google Scholar
Blackadar, B.. K-theory for operator algebras, 2nd ed. Mathematical Sciences Research Institute Publications, 5. Cambridge University Press, Cambridge, 1998.Google Scholar
Bekka, B. and de la Harpe, P.. Unitary representations of groups, duals, and characters. Mathematical Surveys and Monographs, 250. American Mathematical Society, Providence, RI, 2020.Google Scholar
Clare, P.. Hilbert modules associated to parabolically induced representations. J. Operat. Theor. 69 (2) (2013), 483–509.CrossRefGoogle Scholar
Clares, P., Crisp, T., and Higson, N.. Parabolic induction and restriction via C∗-algebras and Hilbert C∗-modules. Compos. Math. 152 (6) (2016), 1286–1318.Google Scholar
Dixmier, J.. C∗-algebras. Translated from the French by Francis Jellett. North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.Google Scholar
Duflo, M. and Moore, C. C.. On the regular representation of a non-unimodular locally compact group. J. Func. Anal. 21 (1976), 209–243.CrossRefGoogle Scholar
Folland, G. B.. A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.Google Scholar
Frank, M. and Larson, D. R.. Frames in Hilbert C∗-modules and C∗-algebras. J. Operat. Theor. 48 (2) (2002), 273–314.Google Scholar
Gomez Aparicio, M. P., Julg, P., and Valette, A.. The Baum-Connes conjecture: An extended survey. In Advances in noncommutative geometry – On the occasion of Alain Connes’ 70th birthday, 127–244, Springer, Cham, 2019.Google Scholar
Green, P.. Square-integrable representations and the dual topology. J. Func. Anat. 35 (1980), 279–294.Google Scholar
Howe, R.. On a notion of rank for unitary representations of the classical groups. Harmonic Analysis and Group Representations, 223–331, Liguori, Naples, 1982.Google Scholar
Kasparov, G. G.. Hilbert C∗-modules: Theorems of Stinespring and Voiculescu, J. Operat. Theor. 4 (1980) 133–150.Google Scholar
Kasparov, G. G.. The operator K-functor and extensions of C∗-algebras. Math. USSR-Izv. 16 (3) (1981), 513–572.CrossRefGoogle Scholar
Kasparov, G. G.. Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91 (1988), 147–201.CrossRefGoogle Scholar
Lance, E. C.. Hilbert C∗-modules, a toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210. Cambridge University Press, Cambridge, 1995.Google Scholar
Li, J.-S.. Singular unitary representations of classical groups. Invent. Math. 97 (2) (1989), 237–255.CrossRefGoogle Scholar
Li, J.-S.. Minimal representations and reductive dual pairs. In Representation theory of Lie groups (Park City, UT, 1998), 293–340, IAS/Park City Mathematics Series, 8, American Mathematical Society, Providence, RI, 2000.Google Scholar
Manuilov, V. M. and Troitsky, E. V.. Hilbert C∗-modules. Translated from the 2001 Russian original by the authors. Translations of Mathematical Monographs, 226. American Mathematical Society, Providence, RI, 2005.Google Scholar
Mesland, B. and S¸ engün, M. H.. Stable range local theta correspondence via C∗-correspondences. Preprint.Google Scholar
Milicˇic’, D.. Topological representation of the group C∗-algebra of SL(2,R). Glasnik Mat. Ser. III 6 (26) (1971), 231–246.Google Scholar
Raeburn, I. and Williams, D. P.. Morita equivalence and continuous trace C∗-algebras. Mathematical Surveys and Monographs, 60. American Mathematical Society, Providence, RI, 1998.Google Scholar
Pierrot, F.. Induction parabolique et K-théorie de C∗-algèbres maximales. C. R. Acad. Sci. Paris Sér. I Math. 332 (9) (2001), 805–808.CrossRefGoogle Scholar
Rieffel, M. A.. Induced representations of C∗-algebras. Adv. Math. 13 (1974), 176–257.CrossRefGoogle Scholar
Rieffel, M. A.. Strong Morita equivalence of certain transformation group C∗-algebras. Math. Ann. 222 (1) (1976), 7–22.CrossRefGoogle Scholar
Rieffel, M. A.. Morita equivalence for operator algebras. In Operator algebras and applications, Part 1 (Kingston, ON, 1980), pp. 285–298, Proceedings of Symposia in Pure Mathematics, 38, American Mathematical Society, Providence, RI, 1982.Google Scholar
Rieffel, M. A.. Applications of strong Morita equivalence to transformation group C∗-algebras. In Operator algebras and applications, Part 1 (Kingston, ON, 1980), 299–310, Proceedings of Symposia in Pure Mathematics, 38, American Mathematical Society, Providence, RI, 1982.Google Scholar
Rørdam, M., Larsen, F., and Lautsten, N.. An introduction to K-theory for C∗-algebras. London Mathematical Society Student Texts, 49. Cambridge University Press, Cambridge, 2000.Google Scholar
Soudry, D.. Explicit Howe duality in the stable range. J. Reine Angew. Math. 396 (1989), 70–86.Google Scholar
Valette, A.. Minimal projections, integrable representations and property (T). Arch. Math. (Basel) 43 (5) (1984), 397–406.CrossRefGoogle Scholar
Wang, S. P.. On isolated points in the dual spaces of locally compact groups. Math. Ann. 218 (1975), 19–34.CrossRefGoogle Scholar