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Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra On

Published online by Cambridge University Press:  12 October 2009

A.L. Carey ,
J. Phillips  and
A. Rennie
A.L. Carey
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia; [email protected]
J. Phillips
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada; [email protected]
A. Rennie
Affiliation:
Australian National University, Canberra, ACT, Australia; [email protected]
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Abstract

This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K1-group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.

Keywords

KMS stateCuntz algebrasindex theory
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 2 , October 2010 , pp. 339 - 380
Copyright
Copyright © ISOPP 2009

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References

Ahl.Ahlfors, L. V., Complex Analysis, McGraw-Hill, 3rd Ed, 1979.Google Scholar
Ar.Araki, H., Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Univ. 11 (1976), pp 809–833 and Relative entropy for states of von Neumann algebras II, Publ. RIMS, Kyoto Univ. 13 (1977), pp 173–192.Google Scholar
APS1.Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral Asymmetry and Riemannian Geometry. I, Math. Proc. Camb. Phil. Soc. 77, (1975), pp 43–69.CrossRefGoogle Scholar
APS3.Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral Asymmetry and Riemannian Geometry. III, Math. Proc. Camb. Phil. Soc. 79, (1976), pp 71–99.CrossRefGoogle Scholar
BPRS.Bates, T., Pask, D., Raeburn, I., Szymanski, W., The C*-Algebras of Row-Finite Graphs, New York J.Math. 6 (2000), pp 307–324.Google Scholar
BR1.Bratteli, O., Robinson, D., Operator Algebras and Quantum Statistical Mechanics 1, Springer-Verlag, 2nd Ed, 1987.CrossRefGoogle Scholar
BR2.Bratteli, O., Robinson, D., Operator Algebras and Quantum Statistical Mechanics 2, Springer-Verlag, 2nd Ed, 1987.CrossRefGoogle Scholar
CP1.Carey, A. L., Phillips, J., Unbounded Fredholm Modules and Spectral Flow, Canadian J. Math. 50 (4) (1998), pp 673–718.CrossRefGoogle Scholar
CP2.Carey, A. L., Phillips, J., Spectral Flow in θ-summable Fredholm Modules, Eta Invariants and the JLO Cocycle, K-Theory 31 (2004), pp 135–194.CrossRefGoogle Scholar
CPR.Carey, A., Phillips, J., Rennie, A., A noncommutative Atiyah-Patodi-Singer index theorem in KK-Theory, to appear in J. Reine Angew. Math.Google Scholar
CRT.Carey, A., Rennie, A., Tong, K., Spectral flow invariants and twisted cyclic theory from the Haar state on SUq(2), to appear in J. Geom. Phys.Google Scholar
CPS2.Carey, A., Phillips, J., Sukochev, F., Spectral Flow and Dixmier Traces, Adv. Math, 173 (2003), pp 68–113.CrossRefGoogle Scholar
CPRS1.Carey, A., Phillips, J., Rennie, A., Sukochev, F., The Hochschild Class of the Chern Character of Semifinite Spectral Triples, Journal of Functional Analysis, 213 (2004), pp 111–153.CrossRefGoogle Scholar
CPRS2.Carey, A., Phillips, J., Rennie, A., Sukochev, F., The local index formula in semifinite von Neumann algebras I: Spectral Flow, Adv. in Math. 202 (2006), pp 451–516.CrossRefGoogle Scholar
C.Connes, A., Noncommutative Geometry, Academic Press, 1994.Google Scholar
C1.Connes, A., Gravity Coupled with Matter and the Foundation of Noncommutative Geometry, Commun. Math. Phys. 182 (1996), pp 155–176.CrossRefGoogle Scholar
CM.Connes, A., Moscovici, H., The Local Index Formula in Noncommutative Geometry, Geom. Funct. Analysis 5 (1995), 174–243.CrossRefGoogle Scholar
CoM.Connes, A., Moscovici, H., Type III and spectral triples, arXiv:math/0609703.Google Scholar
Cu.Cuntz, J., Simple C*-algebras generated by isometries, Commun. Math. Phys, 57 (1977), pp 173–189.CrossRefGoogle Scholar
Dav.Davidson, K., C*-Algebras by Example, Fields Institute Monographs, Amer. Math. Soc. Providence, 1996.CrossRefGoogle Scholar
FK.Fack, T. and Kosaki, H., Generalised s-numbers of τ-measurable operators, Pac. J. Math. 123 (1986), pp 269–300.CrossRefGoogle Scholar
G.Goswami, Debashish, Twisted Entire Cyclic Cohomology, JLO-Cocycles and Equivariant Spectral Triples, Rev. Math. Phys. 16 No. 5 (2004), pp 583–602.CrossRefGoogle Scholar
GVF.Gracia-Bondía, J. M., Varilly, J. C., Figueroa, H., Elements of Noncommutative Geometry, Birkhauser, Boston, 2001.CrossRefGoogle Scholar
HK.Hadfield, T., Krähmer, U., Twisted Homology of Quantum SL(2), K-Theory 34 No. 4 (2005), 327–360CrossRefGoogle Scholar
H.Higson, N., The Local Index Formula in Noncommutative Geometry, Contemporary Developments in Algebraic K-Theory, ictp Lecture Notes 15, (2003), pp 444–536.Google Scholar
HR.Higson, N., Roe, J., Analytic K-Homology, Oxford University Press, 2000.Google Scholar
KNR.Kaad, J., Nest, R., Rennie, A., KK-Theory and spectral flow in von Neumann algebras, arXive:math.OA/0701326.Google Scholar
KR.Kadison, R.V., Ringrose, J. R., Fundamentals of the Theory of Operator Algebras. Vol II Advanced Theory, Academic Press, 1986.Google Scholar
K.Kasparov, G. G., The Operator K-Functor and Extensions of C*-Algebras, Math. USSR. Izv. 16 No. 3 (1981), pp 513–572.CrossRefGoogle Scholar
KPR.Kumjian, A., Pask, D. and Raeburn, I., Cuntz-Krieger Algebras of Directed Graphs, Pac. J. Math. 184 (1998), pp 161–174.CrossRefGoogle Scholar
KMT.Kustermans, J., Murphy, G., Tuset, L., Differential Calculi over Quantum Groups and Twisted Cyclic Cocycles, J. Geom. Phys., 44 (2003), pp 570–594.CrossRefGoogle Scholar
L.Lance, E. C., Hilbert C*-Modules, Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
LSS.Lord, S., Sedaev, A., Sukochev, F. A., Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators, Journal of Functional Analysis, 224 no.1 (2005), pp 72–106.CrossRefGoogle Scholar
PR.Pask, D., Rennie, A., The Noncommutative Geometry of Graph C*-Algebras I: The Index Theorem, Journal of Functional Analysis, 233 (2006), pp 92–134.CrossRefGoogle Scholar
PRS2.Pask, D., Rennie, A., Sims, A., Noncommutative Manifolds from Graph and k-Graph C*-Algebras, to appear in Comm. Math. Phys.Google Scholar
Ped.Pedersen, G. K., C*-algebras and their automorphism groups, London Math. Soc. monographs 14, Academic Press, London 1979.Google Scholar
PT.Pedersen, G. K., Takesaki, M., The Radon-Nikodym Theorem for von Neumann Algebras, Acta Math. 130 (1973), pp 53–87.CrossRefGoogle Scholar
Pu.Putnam, I., An Excision Theorem for the K-Theory of C*-Algebras, J. Operator Theory 38 (1997), pp 151–171.Google Scholar
S.Schweitzer, L. B., A Short Proof that Mn(A) is local if A is Local and Fréchet, Int. J. Math. 3 No.4 (1992), pp 581–589.CrossRefGoogle Scholar
T.Tomoyama, J., On the projection of norm one in W*-algebras, Proc. Japan Acad. 33 (1957), pp 608–612.Google Scholar