On the distinction between the classes of Dixmier and Connes-Dixmier traces

Sukochev, Fedor; Usachev, Alexandr; Zanin, Dmitriy

doi:10.1090/S0002-9939-2012-11853-2

On the distinction between the classes of Dixmier and Connes-Dixmier traces
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by Fedor Sukochev, Alexandr Usachev and Dmitriy Zanin

Proc. Amer. Math. Soc. 141 (2013), 2169-2179

DOI: https://doi.org/10.1090/S0002-9939-2012-11853-2

Published electronically: December 28, 2012

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Abstract:

We prove that the classes of Dixmier and Connes-Dixmier traces differ even on the Dixmier ideal $\mathcal M_{1,\infty }$. We construct a Marcinkiewicz space $\mathcal M_\psi$ and a positive operator $T\in \mathcal M_\psi$ which is Connes-Dixmier measurable but which is not Dixmier measurable.

References

Alan Carey, John Phillips, and Fyodor Sukochev, Spectral flow and Dixmier traces, Adv. Math. 173 (2003), no. 1, 68–113. MR 1954456, DOI 10.1016/S0001-8708(02)00015-4
A. L. Keri and F. A. Sukochev, Dixmier traces and some applications to noncommutative geometry, Uspekhi Mat. Nauk 61 (2006), no. 6(372), 45–110 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 6, 1039–1099. MR 2330013, DOI 10.1070/RM2006v061n06ABEH004369
Alan L. Carey, Adam Rennie, Aleksandr Sedaev, and Fyodor Sukochev, The Dixmier trace and asymptotics of zeta functions, J. Funct. Anal. 249 (2007), no. 2, 253–283. MR 2345333, DOI 10.1016/j.jfa.2007.04.011
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Jacques Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107–A1108 (French). MR 196508
P. G. Dodds, B. de Pagter, A. A. Sedaev, E. M. Semenov, and F. A. Sukochev, Singular symmetric functionals, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290 (2002), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 30, 42–71, 178 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 124 (2004), no. 2, 4867–4885. MR 1942537, DOI 10.1023/B:JOTH.0000042448.87252.0f
P. G. Dodds, B. de Pagter, E. M. Semenov, and F. A. Sukochev, Symmetric functionals and singular traces, Positivity 2 (1998), no. 1, 47–75. MR 1655756, DOI 10.1023/A:1009720826217
N. J. Kalton, A. A. Sedaev, and F. A. Sukochev, Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces, Adv. Math. 226 (2011), no. 4, 3540–3549. MR 2764897, DOI 10.1016/j.aim.2010.09.025
Steven Lord, Aleksandr Sedaev, and Fyodor Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal. 224 (2005), no. 1, 72–106. MR 2139105, DOI 10.1016/j.jfa.2005.01.002
Steven Lord and Fedor Sukochev, Measure theory in noncommutative spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper 072, 36. MR 2725011, DOI 10.3842/SIGMA.2010.072
A. Pietsch, Dixmier Traces of Operators on Banach and Hilbert Spaces, to appear in Math. Nachr.
A. Pietsch, Shift-invariant functionals on Banach sequence spaces, preprint.
A. Pietsch, Connes-Dixmier Traces Versus Dixmier Traces, preprint.
A. Sedaev, Geometrical and topological aspects of interpolation spaces of Petre’s K-method, Thesis, 2010 (in Russian).
F. Sukochev, A. Usachev and D. Zanin, Generalized limits with additional invariance properties and their applications to noncommutative geometry, submitted manuscript.
F. Sukochev and D. Zanin, Traces on symmetrically normed operator ideals, to appear in J. Reine Angew. Math.