Sums of commutators in ideals and modules of type II factors

[Sommes des commutateurs dans les idéaux et les modules de facteurs de type II]

J. Dykema, Kenneth ; J. Kalton, Nigel

Annales de l'Institut Fourier, Tome 55 (2005), p. 931-971 / Harvested from Numdam

Résumé

Soit ${\mathcal {M}}$ un facteur de type II$_\infty $ ou II$_1$ avec prédual séparable et soit $\overline{\mathcal {M}}$ l’algèbre d’operateurs $\tau $-mesurable et affilié avec ${\mathcal {M}}$. Nous caractérisons l’espace des sommes de commutateurs $[{\mathcal {I}},{\mathcal {J}}]$ pour tous sous-$({\mathcal {M}}, {\mathcal {M}})$-bimodules ${\mathcal {I}}$ et ${\mathcal {J}}$ de $\overline{\mathcal {M}}$.

Abstract

Let ${\mathcal {M}}$ be a factor of type II$_\infty $ or II$_1$ having separable predual and let $\overline{\mathcal {M}}$ be the algebra of affiliated $\tau $-measurable operators. We characterize the commutator space $[{\mathcal {I}},{\mathcal {J}}]$ for sub-$({\mathcal {M}},{\mathcal {M}})$- bimodules ${\mathcal {I}}$ and ${\mathcal {J}}$ of $\overline{\mathcal {M}}$.

Détail

Publié le : 2005-01-01

MR 2149407 | Zbl 1090.46046

DOI : https://doi.org/10.5802/aif.2118
Classification:  46L10,  46L52
Mots clés: commutateurs, facteurs de type II, mesure de Brown, l'espace de fonctions non commutatives

Bibliographie

[1] J. H. Anderson Commutators in ideals of trace class operators II, Indiana Univ. Math. J., Tome 35 (1986), pp. 373-378 | Article | MR 833400 | Zbl 0602.47033

[2] L. G. Brown Lidskii's theorem in the type II case, Proc. U. S. Japan Seminar, Kyoto 1983 (Pitman Research Notes Math.) Tome 123 (1986), pp. 1-35 | Zbl 0646.46058

[3] J. W. Calkin Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math., Tome 42 (1941), pp. 839-873 | Article | MR 5790 | Zbl 0063.00692

[4] C. Cecchini On two definitions of measurable and locally measurable operators, Bullettino U. M. I, Tome 15-A (1978) no. 5, pp. 526-534 | MR 521097 | Zbl 0392.46042

[5] P. G. Dodds; T. K. Dodds; B. De Pagter Noncommutative Banach function spaces, Math. Z., Tome 201 (1989), pp. 583-597 | Article | MR 1004176 | Zbl 0653.46061

[6] P. G. Dodds; B. De Pagter; E. M. Semenov; F. A. Sukochev Symmetric functionals and singular traces, Positivity, Tome 2 (1998), pp. 45-75 | MR 1655756 | Zbl 0915.46021

[7] K. J. Dykema; T. Figiel; G. Weiss; M. Wodzicki The commutator structure of operator ideals, Adv. Math., Tome 185 (2004), pp. 1-79 | Article | MR 2058779 | Zbl 02081961

[8] K. J. Dykema; N. J. Kalton Spectral characterization of sums of commutators II, J. reine angew. Math., Tome 504 (1998), pp. 127-137 | MR 1656763 | Zbl 0919.47023

[9] T. Fack Sur la notion de valeur caractéristique, J. Operator Theory, Tome 7 (1982), pp. 307-333 | MR 658616 | Zbl 0493.46052

[10] T. Fack Sums of commutators in non-commutative Banach function spaces, J. Funct. Anal., Tome 207 (2004), pp. 358-398 | Article | MR 2032994 | Zbl 1062.46053

[11] T. Fack; P. De La Harpe Sommes de commutateurs dans les algègres de von Neumann finies continues, Ann. Inst. Fourier, Grenoble, Tome 30 (1980), pp. 49-73 | Article | Numdam | MR 597017 | Zbl 0425.46046

[12] T. Fack; H. Kosaki Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math., Tome 123 (1986), pp. 269-300 | MR 840845 | Zbl 0617.46063

[13] T. Figiel; N. J. Kalton; M. Cwikel, M. Englis, A. Kufner, L.-E. Persson And G. Sparr Symmetric linear functionals on function spaces, Function spaces, interpolation theory and related topics, de Gruyter (2002), pp. 311-332 | Zbl 1032.46043

[14] B. Fuglede; R. Kadison Determinant theory in finite factors, Ann. Math., Tome 55 (1952), pp. 520-530 | Article | MR 52696 | Zbl 0046.33604

[15] A. Grothendieck Séminaire Bourbaki, 1954/55, Paris, Textes des conférences, second corrected edition, Secrétariat mathématique, Tome 113 (1959) | Zbl 0161.33603

[16] D. Guido; T. Isola Singular traces on semifinite von Neumann algebras, J. Funct. Anal., Tome 134 (1995), pp. 451-485 | Article | MR 1363808 | Zbl 0861.46036

[17] H. Halpern Commutators in properly infinite von Neumann algebras, Trans. Amer. Math. Soc., Tome 139 (1969), pp. 55-73 | Article | MR 251546 | Zbl 0176.11403

[18] V. Kaftal; G. Weiss Traces, ideals, and arithmetic means, Proc. Nat. Acad. Sci., Tome 99 (2002), pp. 7356-7360 | Article | MR 1907839 | Zbl 1015.47016

[19] N. J. Kalton Trace-class operators and commutators, J. Funct. Anal., Tome 86 (1989), pp. 41-74 | Article | MR 1013933 | Zbl 0684.47017

[20] N. J. Kalton Spectral characterization of sums of commutators I, J. reine angew. Math., Tome 504 (1998), pp. 115-125 | MR 1656767 | Zbl 0919.47022

[21] N. J. Kalton Traces and characteristic determinants, Rend. Circ. Math. Palermo, Tome 56 (1998), pp. 59-68 | MR 1710822 | Zbl 0940.47020

[22] F. J. Murray; J. Von Neumann On rings of operators, Ann. of Math., Tome 37 (1936), pp. 116-229 | Article | JFM 62.0449.03 | MR 1503275 | Zbl 0014.16101

[23] E. Nelson Notes on non-commutative integration, J. Funct. Anal., Tome 15 (1974), pp. 103-116 | Article | MR 355628 | Zbl 0292.46030

[24] V. I. Ovcinnikov The $s$-numbers of measurable operators, (Russian), Funkcional. Anal. i Prilozen, Tome 4 (1970) no. 017, pp. 78-85 | MR 271763 | Zbl 0219.47029

[25] C. Pearcy; D. Topping On commutators in ideals of compact operators, Michigan J. Math., Tome 18 (1971), pp. 247-252 | Article | MR 284853 | Zbl 0226.46066

[26] I. E. Segal A non-commutative extension of abstract integration, Ann. of Math., Tome 57 (1953), pp. 401-457 | Article | MR 54864 | Zbl 0051.34201

[27] A. Ströh; G. P. West $\tau$-compact operators affiliated to a semifinite von Neumann algebra, Proc. R. Ir. Acad., Tome 93A (1993), pp. 73-86 | MR 1241841 | Zbl 0790.46049

[28] G. Weiss Commutators of Hilbert-Schmidt operators II, Integral Equations Operator Theory, Tome 3 (1980) no. 4, pp. 574-600 | MR 595752 | Zbl 0455.47025

[29] G. P. West Ideals of $\tau$-measurable operators, Quaestiones Math., Tome 18 (1995), pp. 333-344 | Article | MR 1354116 | Zbl 0837.46051