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FREDHOLM MODULES OVER GRAPH $C^{\ast }$-ALGEBRAS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  19 June 2015

TYRONE CRISP
TYRONE CRISP*
Affiliation:
Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen Ø, Denmark email [email protected]
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Abstract

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We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in $K$-homology of graph $C^{\ast }$-algebras. We prove that every $K$-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators, and we exhibit generating Fredholm modules for the $K$-homology of quantum lens spaces.

Keywords

K-homologygraph C∗ -algebras

MSC classification

Primary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 302 - 315
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Arici, F., Brain, S. and Landi, G., ‘The Gysin sequence for quantum lens spaces’, Preprint, 2014, http://arxiv.org/abs/1401.6788.Google Scholar
Arici, F., Kaad, J. and Landi, G., ‘Pimsner algebras and Gysin sequences from principal circle actions’, Preprint, 2014, http://arxiv.org/abs/1409.5335.Google Scholar
Atiyah, M. F., K-Theory, Lecture Notes by D. W. Anderson (W. A. Benjamin, New York, 1967).Google Scholar
Blackadar, B., K-Theory for Operator Algebras, 2nd edn, Mathematical Sciences Research Institute Publications, 5 (Cambridge University Press, Cambridge, 1998).Google Scholar
Crisp, T., ‘Corners of graph algebras’, J. Operator Theory 60(2) (2008), 253–271.Google Scholar
Cuntz, J., ‘A class of C ∗ -algebras and topological Markov chains. II. Reducible chains and the Ext-functor for C ∗ -algebras’, Invent. Math. 63(1) (1981), 25–40.CrossRefGoogle Scholar
Cuntz, J. and Krieger, W., ‘A class of C ∗ -algebras and topological Markov chains’, Invent. Math. 56(3) (1980), 251–268.CrossRefGoogle Scholar
Drinen, D. and Tomforde, M., ‘Computing K-theory and Ext for graph C ∗ -algebras’, Illinois J. Math. 46(1) (2002), 81–91.CrossRefGoogle Scholar
Goffeng, M. and Mesland, B., ‘Spectral triples and finite summability on Cuntz–Krieger algebras’, Preprint, 2014, http://arxiv.org/abs/1401.2123.CrossRefGoogle Scholar
Hawkins, E. and Landi, G., ‘Fredholm modules for quantum Euclidean spheres’, J. Geom. Phys. 49(3–4) (2004), 272–293.CrossRefGoogle Scholar
Higson, N. and Roe, J., Analytic K-Homology, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000).Google Scholar
Hong, J. H. and Szymański, W., ‘Quantum spheres and projective spaces as graph algebras’, Comm. Math. Phys. 232(1) (2002), 157–188.CrossRefGoogle Scholar
Hong, J. H. and Szymański, W., ‘Quantum lens spaces and graph algebras’, Pacific J. Math. 211(2) (2003), 249–263.CrossRefGoogle Scholar
Hong, J. H. and Szymański, W., ‘Noncommutative balls and mirror quantum spheres’, J. Lond. Math. Soc. (2) 77(3) (2008), 607–626.CrossRefGoogle Scholar
Kumjian, A. and Pask, D., ‘C ∗ -algebras of directed graphs and group actions’, Ergod. Th. & Dynam. Sys. 19(6) (1999), 1503–1519.CrossRefGoogle Scholar
Lance, E. C., ‘The compact quantum group SO(3)q’, J. Operator Theory 40(2) (1998), 295–307.Google Scholar
Pask, D. and Raeburn, I., ‘On the K-theory of Cuntz–Krieger algebras’, Publ. Res. Inst. Math. Sci. 32(3) (1996), 415–443.CrossRefGoogle Scholar
Podleś, P., ‘Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups’, Comm. Math. Phys. 170(1) (1995), 1–20.CrossRefGoogle Scholar
Raeburn, I., Graph Algebras, CBMS Regional Conference Series in Mathematics, 103 (American Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
Raeburn, I. and Szymański, W., ‘Cuntz–Krieger algebras of infinite graphs and matrices’, Trans. Amer. Math. Soc. 356(1) (2004), 39–59 (electronic).CrossRefGoogle Scholar
Rosenberg, J. and Schochet, C., ‘The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor’, Duke Math. J. 55(2) (1987), 431–474.CrossRefGoogle Scholar
Szymański, W., ‘General Cuntz–Krieger uniqueness theorem’, Internat. J. Math. 13(5) (2002), 549–555.CrossRefGoogle Scholar
Tomforde, M., ‘Computing Ext for graph algebras’, J. Operator Theory 49(2) (2003), 363–387.Google Scholar
Vaksman, L. L. and Soĭbel’man, Ya. S., ‘Algebra of functions on the quantum group SU(n + 1), and odd-dimensional quantum spheres’, Algebra i Analiz 2(5) (1990), 101–120.Google Scholar
Woronowicz, S. L., ‘Twisted SU(2) group. An example of a noncommutative differential calculus’, Publ. Res. Inst. Math. Sci. 23(1) (1987), 117–181.CrossRefGoogle Scholar
Yi, I., ‘K-theory and K-homology of C ∗ -algebras for row-finite graphs’, Rocky Mountain J. Math. 37(5) (2007), 1723–1742.CrossRefGoogle Scholar