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K-THEORY FOR THE C*-ALGEBRAS OF THE SOLVABLE BAUMSLAG–SOLITAR GROUPS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  18 October 2017

SANAZ POOYA  and
ALAIN VALETTE
SANAZ POOYA
Affiliation:
Institut de Mathéematiques, Université de Neuchâtel Unimail, Rue Emile Argand 11 CH-2000 Neuchatel, Switzerland e-mail: [email protected], [email protected]
ALAIN VALETTE
Affiliation:
Institut de Mathéematiques, Université de Neuchâtel Unimail, Rue Emile Argand 11 CH-2000 Neuchatel, Switzerland e-mail: [email protected], [email protected]
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Abstract

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We provide a new computation of the K-theory of the group C*-algebra of the solvable Baumslag–Solitar group BS(1, n) (n ≠ 1); our computation is based on the Pimsner–Voiculescu 6-terms exact sequence, by viewing BS(1, n) as a semi-direct product ℤ[1/n] ⋊ ℤ. We deduce from it a new proof of the Baum–Connes conjecture with trivial coefficients for BS(1, n).

MSC classification

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 2 , May 2018 , pp. 481 - 486
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Béguin, C., Bettaieb, H. and Valette, A., K-theory for the C*-algebras of one-relator groups, K-Theory 16 (1999), 277–298.Google Scholar
2. Baum, P. and Connes, A., Geometric K-theory for Lie groups and foliations, Enseign. Math. 46 (2000), 3–42.Google Scholar
3. Baum, P., Connes, A. and Higson, N., Classifying space for proper actions and K-theory of group C*-algebras, Contemp. Math. 164 (1994), 241–292.Google Scholar
4. Blackadar, B., K-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5 (Springer-Verlag, New York, 1986).CrossRefGoogle Scholar
5. Brenken, B., K-groups of solenoidal algebras I, Proc. Amer. Math. Soc. 123 (5) (1995), 1457–1464.Google Scholar
6. Brenken, B. and Jørgensen, P. E. T., A family of dilation crossed product algebras, J. Oper. Theory 25 (1991), 299–308.Google Scholar
7. Higson, N. and Kasparov, G., E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), 23–74.Google Scholar
8. Iochum, B., Masson, T., Schücker, T. and Sitarz, A., κ-Deformation and spectral triples, Acta Phys. Polon. Supp. 4 (2011), 305–324.CrossRefGoogle Scholar
9. Lyndon, R. C., Cohomology theory of groups with a single defining relation, Ann. Math. 52 (1950), 650–665.Google Scholar
10. Mislin, G. and Valette, A., Proper group actions and the Baum–Connes conjecture. Advanced courses in mathematics - CRM Barcelona (Birkhäuser, Basel, 2003).Google Scholar
11. Oyono-Oyono, H., La conjecture de Baum–Connes pour les groupes agissant sur les arbes, C. R. Acad. Sci. Paris, Sér. I 326 (1998), 799–804.Google Scholar
12. Tu, J.-L., The Baum–Connes conjecture and discrete group actions on trees, K-Theory 17 (1999), 303–318.Google Scholar
13. Pimsner, M. and Voiculescu, D., Exact sequences for K-groups and Ext-groups of certain cross product C*- algebras, Oper. Theory 4 (1980), 93–118.Google Scholar
14. Sudo, T., K-theory for the group C*-algebras of certain solvable discrete groups, Hokkaido Math. J. 43 (2014), 209–260.CrossRefGoogle Scholar