Abstract
Using Harish-Chandra parameter space , an explicit formula for the K-theory of the reduced \(C^*\)-algebra of \(\mathrm{GL}(n,{\mathbb {C}})\) is obtained, in analogy with the real case \(\mathrm{GL}(n,{\mathbb {R}})\) [8]. Applying automorphic induction, an instance of Langlands functoriality principle, we then relate the K-theory of \(C_r^*\mathrm{GL}(2n,{\mathbb {R}})\) and \(C_r^*\mathrm{GL}(n,{\mathbb {C}})\).
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Echterhoff, S., Pfante, O.: Equivariant \(K\)-theory of finite-dimensional real vector spaces. Munster J. of Math. 2, 65–94 (2009)
Harish-Chandra: Collected Papers, vol. 4. Springer, Berlin (1984)
Henniart, G.: Induction automorphe pour \({\rm {GL}}(n,{\mathbb{C}})\). J. Funct. Anal. 258(9), 3082–3096 (2010)
Khalkhali, M.: Basic Noncommutative Geometry, 2nd edn. EMS, Zürich (2013)
Knapp, A.: Local Langlands correspondence: the archimedean case. Proc. Symp. Pure. Math. 55, 393–410 (1994)
Plymen, R.: The reduced \(C^*\)-algebra of the p-adic group \({\rm {GL}}(n)\). J. Funct. Anal. 72, 1–12 (1987)
Mendes, S., Plymen, R.: Base change and \(K\)-theory for \({\rm{GL}}(n)\). J. Noncommut. Geom. 1, 311–331 (2007)
Mendes, S., Plymen, R.: Functoriality and \(K\)-theory for \({\rm {GL}}_n({\mathbb{R}})\). arXiv Preprint (2015) (To appear in the Münster Journal of Mathematics)
Penington, M., Plymen, R.: The Dirac operator and the principal series for complex semisimple Lie groups. J. Funct. Anal. 53, 269–286 (1983)
Acknowledgments
I would like to thank the organizers of the Mattriad’2015 held in Coimbra, specially Professors Natália Bebiano, Cristina Câmara and Ana Nata. I also thank the anonymous referee whose remarks helped to improve the paper.
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Mendes, S. (2017). On the K-Theory of the Reduced \(C^*\)-Algebras of \(GL(n,\mathbb {R})\) and \(GL(n,\mathbb {C})\) . In: Bebiano, N. (eds) Applied and Computational Matrix Analysis. MAT-TRIAD 2015. Springer Proceedings in Mathematics & Statistics, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-49984-0_6
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DOI: https://doi.org/10.1007/978-3-319-49984-0_6
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