Geodesic
On Cardinal Invariants and Generators for von Neumann Algebras
Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 455-480

Voir la notice de l'article provenant de la source Cambridge University Press

We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal{M}$ can be computed from the decomposability number, $\text{dens}\left( \mathcal{M} \right)$ , and the minimal cardinality of a generating set, $\text{gen}\left( \mathcal{M} \right)$ . Applications include the equivalence of the well-known generator problem, “Is every separably-acting von Neumann algebra singly-generated?”, with the formally stronger questions, “Is every countably-generated von Neumann algebra singly-generated?” and “Is the gen invariant monotone?” Modulo the generator problem, we determine the range of the invariant $\left( \text{gen}\left( \mathcal{M} \right),\,\text{dens}\left( \mathcal{M} \right) \right)$ , which is mostly governed by the inequality $\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$ .
DOI : 10.4153/CJM-2011-048-2
Mots-clés : 46L10, von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation density 
Sherman, David. On Cardinal Invariants and Generators for von Neumann Algebras. Canadian journal of mathematics, Tome 64 (2012) no. 2, pp. 455-480. doi: 10.4153/CJM-2011-048-2
@article{10_4153_CJM_2011_048_2,
     author = {Sherman, David},
     title = {On {Cardinal} {Invariants} and {Generators} for von {Neumann} {Algebras}},
     journal = {Canadian journal of mathematics},
     pages = {455--480},
     year = {2012},
     volume = {64},
     number = {2},
     doi = {10.4153/CJM-2011-048-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-048-2/}
}
TY  - JOUR
AU  - Sherman, David
TI  - On Cardinal Invariants and Generators for von Neumann Algebras
JO  - Canadian journal of mathematics
PY  - 2012
SP  - 455
EP  - 480
VL  - 64
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-048-2/
DO  - 10.4153/CJM-2011-048-2
ID  - 10_4153_CJM_2011_048_2
ER  - 
%0 Journal Article
%A Sherman, David
%T On Cardinal Invariants and Generators for von Neumann Algebras
%J Canadian journal of mathematics
%D 2012
%P 455-480
%V 64
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2011-048-2/
%R 10.4153/CJM-2011-048-2
%F 10_4153_CJM_2011_048_2

[1] [1] Akemann, C. A. and Anderson, J. Lyapunov theorems for operator algebras. Mem. Amer. Math. Soc. 94(1991). Google Scholar

[2] [2] Behncke, H., Generators of finite W*-algebras. Tôhoku Math. J. 24(1972), 401–408. Google Scholar | DOI

[3] [3] Behncke, H., Topics in C*-and von Neumann algebras. In: Lectures on Operator Algebras (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. II), Lecture Notes in Math. 247, Springer, Berlin, 1972, 1–54. Google Scholar

[4] [4] Blackadar, B., Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. In: Encyclopaedia of Mathematical Sciences 122, Operator Algebras and Non-commutative Geometry III, Springer-Verlag, Berlin, 2006. Google Scholar

[5] [5] Comfort, W. W., A survey of cardinal invariants. General Topology and Appl. 1(1971), 163–199. Google Scholar | DOI

[6] [6] Comfort, W. W. and Itzkowitz, G. L., Density character in topological groups. Math. Ann. 226(1977), 223–227. Google Scholar | DOI

[7] [7] Dixmier, J., Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann). Second edition, Gauthier-Villars, Paris, 1969. Google Scholar

[8] [8] Dykema, K., Sinclair, A. Smith, R. and S.White, Generators of II1 factors. Oper. Matrices 2(2008), 555–582. Google Scholar

[9] [9] Elliott, G. A., On approximately finite-dimensional von Neumann algebras. Math. Scand. 39(1976), 91–101. Google Scholar

[10] [10] Farah, I. and Katsura, T. Nonseparable UHF algebras I: Dixmier's problem. Adv. Math., to appear. Google Scholar | DOI

[11] [11] Feldman, J., Nonseparability of certain finite factors. Proc. Amer. Math. Soc. 7(1956), 23–26. Google Scholar | DOI

[12] [12] Ge, L. and Popa, S. On some decomposition properties for factors of type II1. Duke Math. J. 94(1998), 79–101. Google Scholar | DOI

[13] [13] Ge, L. and Shen, J. On the generator problem of von Neumann algebras. In: Third International Congress of Chinese Mathematicians, Part 1, 2, AMS/IP Stud. Adv. Math. 42, pt. 1, 2, Amer. Math. Soc., Providence, RI, 2008, 357–375. Google Scholar

[14] [14] Haagerup, U., The standard form of von Neumann algebras. Math. Scand. 37 (1975), 271–283. Google Scholar

[15] [15] Hewitt, E., A remark on density characters. Bull. Amer. Math. Soc. 52(1946), 641–643. Google Scholar | DOI

[16] [16] Hu, Z., Maximally decomposable von Neumann algebras on locally compact groups and duality. Houston J. Math. 31(2005), 857–881. Google Scholar

[17] [17] Hu, Z. and Neufang, M. Decomposability of von Neumann algebras and the Mazur property of higher level. Canad. J. Math. 58(2006), 768–795. Google Scholar | DOI

[18] [18] Jech, T., Set Theory. Third edition, Springer-Verlag, Berlin, 2003. Google Scholar

[19] [19] Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras II. Graduate Studies in Mathematics 16, Amer. Math. Soc., Providence, 1997. Google Scholar

[20] [20] Kaplansky, I., Quelques résultats sur les anneaux d’opérateurs. C. R. Acad. Sci. Paris 231(1950), 485–486. Google Scholar

[21] [21] Kehlet, E. T., Disintegration theory on a constant field of nonseparable Hilbert spaces. Math. Scand. 43(1978), 353–362. Google Scholar

[22] [22] Kruse, A. H., Badly incomplete normed linear spaces. Math. Z. 83(1964), 314–320. Google Scholar | DOI

[23] [23] Marczewski, E., Séparabilité et multiplication cartésienne des espaces topologiques. Fund. Math. 34(1947), 127–143. Google Scholar

[24] [24] Murray, F. J. and von Neumann, J., On rings of operators IV. Ann. of Math. (2) 44(1943), 716–808. Google Scholar | DOI

[25] [25] Neufang, M., On Mazur's property and property (X). J. Operator Theory 60(2008), 301–316. Google Scholar

[26] [26] von Neumann, J., Zur Algebra der Funktionaloperatoren und Theorie der normalen Operatoren. Math. Ann. 102(1929), 370–427. Google Scholar

[27] [27] Pearcy, C., W*-algebras with a single generator. Proc. Amer. Math. Soc. 13(1962), 831–832. Google Scholar

[28] [28] Pearcy, C., On certain von Neumann algebras which are generated by partial isometries. Proc. Amer. Math. Soc. 15(1964), 393–395. Google Scholar | DOI

[29] [29] Pedersen, G. K., Applications of weak semicontinuity in C*-algebra theory. Duke Math. J. 39(1972), 431–450. Google Scholar | DOI

[30] [30] Pedersen, G. K., C*-Algebras and Their Automorphism Groups. London Mathematical Society Monographs 14, Academic Press, London–New York, 1979. Google Scholar

[31] [31] Pondiczery, E. S., Power problems in abstract spaces. Duke Math. J. 11(1944), 835–837. Google Scholar | DOI

[32] [32] Popa, S., On a problem of Kadison on R. V. maximal abelian C-subalgebras in factors. Invent. Math. 65(1981/82), 269–281. Google Scholar | DOI

[33] [33] Powers, R. T., Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. of Math. (2) 86(1967), 138–171. Google Scholar | DOI

[34] [34] Raynaud, Y., On ultrapowers of non commutative Lp spaces. J. Operator Theory 48(2002), 41–68. Google Scholar

[35] [35] Rickart, C. E., General Theory of Banach Algebras. D. van Nostrand, New York, 1960. Google Scholar

[36] [36] Saitô, T., Generations of von Neumann algebras. In: Lectures on Operator Algebras (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. II), Lecture Notes in Math. 247, Springer, Berlin, 1972, 435–531. Google Scholar

[37] [37] Sakai, S., C*-Algebras and W*-Algebras. Springer–Verlag, New York, 1971. Google Scholar

[38] [38] Shen, J., Type II1 factors with a single generator. J. Operator Theory 62(2009), 421–438. Google Scholar

[39] [39] Sherman, D., On the dimension theory of von Neumann algebras. Math. Scand. 101(2007), 123–147. Google Scholar

[40] [40] Smith, R. R. and Sinclair, A. M., Hochschild Cohomology of Von Neumann Algebras. London Mathematical Society Lecture Note Series 203, Cambridge University Press, Cambridge, 1995. Google Scholar

[41] [41] Smith, R. R. and Sinclair, A. M., Finite von Neumann Algebras and Masas. London Mathematical Society Lecture Note Series 351, Cambridge University Press, Cambridge, 2008. Google Scholar

[42] [42] Suzuki, N. and Saitô, T., On the operators which generate continuous von Neumann algebras. Tôhoku Math. J. 15(1963), 277–280. Google Scholar | DOI

[43] [43] Voiculescu, D., Circular and semicircular systems and free product factors. In: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math. 92, Birkhäuser, Boston, 1990, 45–60. Google Scholar

[44] [44] Weaver, N., Mathematical Quantization. Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, 2001. Google Scholar

[45] [45] Willig, P., On hyperfinite W*-algebras. Proc. Amer. Math. Soc. 40(1973), 120–122. Google Scholar

[46] [46] Willig, P., Generators and direct integral decompositions of W*-algebras. Tôhoku Math. J. 26(1974), 35–37. Google Scholar | DOI

[47] [47] Wogen, W., On generators for von Neumann algebras. Bull. Amer. Math. Soc. 75(1969), 95–99. Google Scholar | DOI

[48] [48] Yamagami, S., Notes on operator categories. J. Math. Soc. Japan 59(2007), 541–555. http://dx.doi.org/10.2969/jmsj/05920541 Google Scholar

Cité par Sources :