Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T17:34:31.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Operator Integrals, Spectral Shift, and Spectral Flow

Published online by Cambridge University Press:  20 November 2018

N. A. Azamov ,
A. L. Carey ,
P. G. Dodds  and
F. A. Sukochev
N. A. Azamov
Affiliation:
(Azamov, Dodds) School of Computer Science, Engineering and Mathematics, Flinders University of South Australia, Bedford Park, 5042, SA, Australia, azam0001@csem.flinders.edu.au, peter@csem.flinders.edu.au
P. G. Dodds
Affiliation:
(Carey)Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia, acarey@maths.anu.edu.au
F. A. Sukochev
Affiliation:
(Sukochev) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia, f.sukochev@unsw.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

Keywords

47A5647B4947A5546L51
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 2 , 01 April 2009 , pp. 241 - 263
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Atiyah, M. F., Patodi, V. K., and Singer, I. M., Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc. 79(1976), no. 1, 71–99.Google Scholar
[2] Azamov, N. A., Dodds, P. G., and Sukochev, F. A., The Krein spectral shift function in semifinite von Neumann algebras. Integral Equations Operator Theory 55(2006), no. 3, 347–362.Google Scholar
[3] Birman, M. Sh. and Pushnitski, A. B., Spectral shift function, amazing and multifaceted. Dedicated to the memory of Mark Grigorievich Krein (1907–1989). Integral Equations Operator Theory 30(1998), no. 2, 191–199.Google Scholar
[4] Birman, M. Sh. and Solomyak, M. Z., Remarks on the spectral shift function. J. Soviet Math.3(1975), 408–419.Google Scholar
[5] Birman, M. Sh. and Solomyak, M. Z., Double Stieltjes operator integrals. I. Izdat. Leningrad Univ., Leningrad, 1973, 27–53.Google Scholar
[6] Bratteli, O. and Robinson, D., Operator algebras and quantum statistical mechanics. I. Springer-Verlag, New York-Berlin, 1979.Google Scholar
[7] Breuer, M., Fredholm theories in von Neumann algebras. I. Math. Ann. 178(1968), 243–254.Google Scholar
[8] Breuer, M., Fredholm theories in von Neumann algebras. II. Math. Ann. 180(1969), 313–325.Google Scholar
[9] Carey, A. L. and Phillips, J., Spectral flow in Fredholm modules, eta invariants and the JLO cocycle. K -Theory 31(2004), no. 2, 135–194.Google Scholar
[10] Carey, A. L., Phillips, J., Rennie, A., and Sukochev, F. A., The Hochschild class of the Chern character of semifinite spectral triples. J. Funct. Anal. 213(2004), no. 1, 111–153.Google Scholar
[11] Carey, A. L., Phillips, J., Rennie, A., and Sukochev, F. A., The local index formula in semifinite von Neumann algebras I. Spectral flow. Adv. Math. 202(2006), no. 2, 451–516.Google Scholar
[12] Carey, A. L., Phillips, J., Rennie, A., and Sukochev, F. A., The local index formula in semifinite von Neumann algebras I. The even case. Adv. Math. 202(2006), no. 2, 517–554.Google Scholar
[13] Carey, A. L., Phillips, J., and Sukochev, F. A., Spectral flow and Dixmier traces. Adv. Math. 173(2003), no. 1, 68–113.Google Scholar
[14] Connes, A., Noncommutative Geometry. Academic Press, San Diego, CA, 1994.Google Scholar
[15] Connes, A. and Moscovici, H., The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(1995), no. 2, 174–243.Google Scholar
[16] Daletskii, Y. L. and Krein, S. G., Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations. Voronež. Gos. Univ. Trudy Sem. Funkcional. Anal. 1956(1956), no. 1, 81–105.Google Scholar
[17] Dixmier, J., von Neumann Algebras. North-Holland Mathematical Library 27, North-Holland Publishing Co., Amsterdam-New York, 1981.Google Scholar
[18] Dodds, P. G., Dodds, T. K., and de Pagter, B., Noncommutative Köthe duality. Trans. Amer. Math. Soc. 339(1993), no. 2, 717–750.Google Scholar
[19] Dodds, P. G., Dodds, T. K., de Pagter, B., and Sukochev, F. A., Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces. J. Funct. Anal. 148(1997), no. 1, 28–69.Google Scholar
[20] Dunford, N. and Schwartz, J. T., Linear Operators, Part I: General theory. John Wiley and Sons, Inc., New York, 1988.Google Scholar
[21] Fack, T. and Kosaki, H., Generalised s-numbers of τ-measurable operators. Pacific J. Math. 123(1986), no. 2, 269–300.Google Scholar
[22] Gelfand, I. M. and Shilov, G. E., Generalized functions. Vol. 1. Properties and operations. Academic Press, New York-London, 1964.Google Scholar
[23] Getzler, E., The odd Chern character in cyclic homology and spectral flow. Topology 32(1993), no. 3, 489–507.Google Scholar
[24] Jacobs, K., Measure and integral. Probability and mathematical statistics. Academic Press, New York-London, 1978.Google Scholar
[25] Krein, M. G., On the trace formula in perturbation theory. Mat. Sbornik N. S. 33(75)(1953), 597–626.Google Scholar
[26] Krein, M. G., Some new studies in the theory of perturbations of self-adjoint operators. First Math. Summer School, Part I, Izdat Naukova Dumka, Kiev, 1964, pp. 103–187.Google Scholar
[27] Krein, S. G., Petunin, J. I., and Semenov, E. M., Interpolation of linear operators. Translations of Mathematical Monographs 54, American Mathematical Society, Providence, RI, 1982.Google Scholar
[28] Lifshits, I. M., On a problem in the theory of perturbations connected with quantum statistics. Uspekhi Mat. Nauk 7(1952), 171–180.Google Scholar
[29] de Pagter, B. and Sukochev, F. A., Differentiation of operator functions in non-commutative Lp-spaces. J. Funct. Anal. 212(2004), no. 1, 28–75.Google Scholar
[30] de Pagter, B., Witvliet, H.,, and Sukochev, F. A., Double operator integrals. J. Funct. Anal. 192(2002), no. 1, 52–111.Google Scholar
[31] Pavlov, B. S., Multidimensional operator integrals. Problems of Math. Anal. 2: Linear operators and operator equations, Izdat. Leningrad. Univ., Leninegrad, 1969, pp. 99–122.Google Scholar
[32] Peller, V. V., Multiple operator integrals and higher operator derivatives. J. Funct. Anal. 233(2006), no. 2, 515–544.Google Scholar
[33] Phillips, J., Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull. 39(1996), no. 4, 460–467.Google Scholar
[34] Phillips, J., Spectral flow in type I and type II factors—a new approach. In: Cyclic cohomology and noncommutative geometry, Fields. Inst. Commun. 17, American Mathematical Society, Providence, RI, 1997, pp. 137–153.Google Scholar
[35] Phillips, J. and Raeburn, I., An index theorem for Toeplitz operators with noncommutative symbol space. J. Funct. Anal. 120(1994), no. 2, 239–263.Google Scholar
[36] Reed, M. and Simon, B., Methods of modern mathematical physics: 1. Functional analysis. Academic Press, New York, 1972.Google Scholar
[37] Reiter, H., Classical harmonic analysis and locally compact groups. Clarendon Press, Oxford, 1968.Google Scholar
[38] Schwartz, J. T., Nonlinear Functional Analysis. Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969.Google Scholar
[39] Solomyak, M. Z. and Stenkin, V. V., On a class of multiple operator Stieltjes integrals. Problems of mathematical analysis, Leningrad University Publ. 2, 1969, 122–134.Google Scholar
[40] Stenkin, V. V.,Multiple operator integrals. Izv. Vyss. Ucebn. Zaved. Matematika 4(1977), 102–115.Google Scholar
[41] Vakhaniya, N. N., Tarieladze, V. I., and Chobanyan, S. A., Probability distributions in Banach spaces. Nauka. Moscow, 1985.Google Scholar
[42] Widom, H.,,When are differentiable functions are differentiable? In: Linear and Complex Analysis Problem Book 3, Part 1, Lecture Notes in Mathematics 1573, Springer-Verlag, Berlin-Heidelberg-New York, 1994, pp. 266–271.Google Scholar
[43] Yosida, K., Functional analysis. Fundamental Principles of Mathematical Sciences 123, Springer-Verlag, Berlin-New York, 1980.Google Scholar