The Kadison-Singer problem. (Le problème de Kadison-Singer.) (French) Zbl 1357.46055
The paper under review is a survey paper, written in French, concerning the famous Kadison-Singer problem. This problem, which was proposed in [Am. J. Math. 81, 383–400 (1959; Zbl 0086.09704)] by R. V. Kadison and I. M. Singer, asks whether every pure state on the algebra \(\mathcal{D}(\ell^{2})\) of diagonal operators on the Hilbert space \(\ell^{2}\) can be uniquely extended to a state on the algebra \(\mathcal{B} (\ell^{2})\) of all bounded operators on \(\ell^{2}\). This problem admits a striking variety of equivalent formulations, pertaining to domains which are sometimes quite far away from the initial context of the Kadison-Singer problem, namely \(C^{*}\)-algebras. The problem was solved in 2013 by A. W. Marcus et al. [Ann. Math. (2) 182, No. 1, 327–350 (2015; Zbl 1332.46056), arXiv:1306.3969], using relatively elementary methods in an extremely ingenious way.
The aim of this survey is not to present the proof of [loc.cit.]– the reader is referred instead to some other references, like for instance [A. Valette, Astérisque 367–368, 451–476, Exp. No. 1088 (2015; Zbl 1356.46002)]. The author rather takes the reader to a tour into the Kadison-Singer problem itself and its various reformulations, leaving him/her at the door of the solution at the end. The text, being issued from notes given at a spring school, is aimed at students, and written in an extremely clear and challenging way. Tools on \(C^{*}\)-algebras, filters, ultrafilters, etc., necessary in order to understand the Kadison-Singer problem are presented in detail. Compressibility, paving of bounded operators on \(\ell^{2}\) and the Dixmier property are dealt with, and some non-trivial equivalent reformulations of the Kadison-Singer conjecture are given. Relations with the Feichtinger conjecture and the Bourgain-Tzafriri conjecture are also presented.
The aim of this survey is not to present the proof of [loc.cit.]– the reader is referred instead to some other references, like for instance [A. Valette, Astérisque 367–368, 451–476, Exp. No. 1088 (2015; Zbl 1356.46002)]. The author rather takes the reader to a tour into the Kadison-Singer problem itself and its various reformulations, leaving him/her at the door of the solution at the end. The text, being issued from notes given at a spring school, is aimed at students, and written in an extremely clear and challenging way. Tools on \(C^{*}\)-algebras, filters, ultrafilters, etc., necessary in order to understand the Kadison-Singer problem are presented in detail. Compressibility, paving of bounded operators on \(\ell^{2}\) and the Dixmier property are dealt with, and some non-trivial equivalent reformulations of the Kadison-Singer conjecture are given. Relations with the Feichtinger conjecture and the Bourgain-Tzafriri conjecture are also presented.
Reviewer: Sophie Grivaux (Amiens)
MSC:
| 46L30 | States of selfadjoint operator algebras |
| 46L05 | General theory of \(C^*\)-algebras |
| 42C15 | General harmonic expansions, frames |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
| 46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |
References:
| [1] | Akemann, C. A.; Paulsen, V., State extensions and the Kadison-Singer problem (2008) |
| [2] | Akemann, Charles A.; Anderson, Joel, Lyapunov theorems for operator algebras, Mem. Amer. Math. Soc., 94, 458, iv+88 p. pp. (1991) · Zbl 0769.46036 · doi:10.1090/memo/0458 |
| [3] | Anderson, Joel, Extensions, restrictions, and representations of states on \(C^{\ast } \)-algebras, Trans. Amer. Math. Soc., 249, 2, 303-329 (1979) · Zbl 0408.46049 · doi:10.2307/1998793 |
| [4] | Anderson, Joel, Extreme points in sets of positive linear maps on \({\mathcal{B}}({\mathcal{H}})\), J. Funct. Anal., 31, 2, 195-217 (1979) · Zbl 0422.46049 · doi:10.1016/0022-1236(79)90061-2 |
| [5] | Arveson, William, An invitation to \(C^*\)-algebras, x+106 p. pp. (1976), Springer-Verlag, New York-Heidelberg · Zbl 0344.46123 |
| [6] | Balan, Radu; Casazza, Peter G.; Heil, Christopher; Landau, Zeph, Density, overcompleteness, and localization of frames. I. Theory, J. Fourier Anal. Appl., 12, 2, 105-143 (2006) · Zbl 1096.42014 · doi:10.1007/s00041-006-6022-0 |
| [7] | Baranov, Anton; Dyakonov, Konstantin, The Feichtinger conjecture for reproducing kernels in model subspaces, J. Geom. Anal., 21, 2, 276-287 (2011) · Zbl 1230.30039 · doi:10.1007/s12220-010-9147-y |
| [8] | Berman, Kenneth; Halpern, Herbert; Kaftal, Victor; Weiss, Gary, Matrix norm inequalities and the relative Dixmier property, Integral Equations Operator Theory, 11, 1, 28-48 (1988) · Zbl 0647.47007 · doi:10.1007/BF01236652 |
| [9] | Bice, Tristan Matthew, Filters in \(\rm C^*\)-algebras, Canad. J. Math., 65, 3, 485-509 (2013) · Zbl 1275.46045 · doi:10.4153/CJM-2011-095-4 |
| [10] | Bøe, Bjarte, An interpolation theorem for Hilbert spaces with Nevanlinna-Pick kernel, Proc. Amer. Math. Soc., 133, 7, 2077-2081 (electronic) (2005) · Zbl 1065.30053 · doi:10.1090/S0002-9939-05-07722-1 |
| [11] | Bourbaki, Nicolas, Espaces vectoriels topologiques. Chapitres 1 à 5, vii+368 p. pp. (1981), Masson, Paris · Zbl 1106.46003 |
| [12] | Bourgain, J.; Tzafriri, L., Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math., 57, 2, 137-224 (1987) · Zbl 0631.46017 · doi:10.1007/BF02772174 |
| [13] | Bourgain, J.; Tzafriri, L., On a problem of Kadison and Singer, J. Reine Angew. Math., 420, 1-43 (1991) · Zbl 0729.47028 |
| [14] | Bownik, Marcin; Speegle, Darrin, The Feichtinger conjecture for wavelet frames, Gabor frames and frames of translates, Canad. J. Math., 58, 6, 1121-1143 (2006) · Zbl 1176.42015 · doi:10.4153/CJM-2006-041-3 |
| [15] | Casazza, Peter G.; Christensen, Ole; Kalton, Nigel J., Frames of translates, Collect. Math., 52, 1, 35-54 (2001) · Zbl 1029.42029 |
| [16] | Casazza, Peter G.; Christensen, Ole; Lindner, Alexander M.; Vershynin, Roman, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc., 133, 4, 1025-1033 (electronic) (2005) · Zbl 1082.46018 · doi:10.1090/S0002-9939-04-07594-X |
| [17] | Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric, Operator theory, operator algebras, and applications, 414, The Kadison-Singer problem in mathematics and engineering : a detailed account, 299-355 (2006), Amer. Math. Soc., Providence, RI · Zbl 1110.46038 · doi:10.1090/conm/414/07820 |
| [18] | Casazza, Peter G.; Peterson, Jesse, An elementary, illustrative proof of the Rado-Horn theorem, Linear Algebra Appl., 437, 10, 2523-2537 (2012) · Zbl 1253.15003 · doi:10.1016/j.laa.2012.06.017 |
| [19] | Chalendar, I.; Fricain, E.; Timotin, D., A short note on the Feichtinger conjecture (2011) |
| [20] | Choquet, Gustave, Deux classes remarquables d’ultrafiltres sur N, Bull. Sci. Math. (2), 92, 143-153 (1968) · Zbl 0162.26201 |
| [21] | Dirac, P. A. M., The Principles of Quantum Mechanics (1930), Oxford, at the Clarendon Press · JFM 56.0745.05 |
| [22] | Dixmier, Jacques, Les \(C^{\ast } \)-algèbres et leurs représentations, xv+390 p. pp. (1969), Gauthier-Villars Éditeur, Paris · Zbl 0152.32902 |
| [23] | Douglas, Ronald G., Banach algebra techniques in operator theory, 179, xvi+194 p. pp. (1998), Springer-Verlag, New York · Zbl 0920.47001 · doi:10.1007/978-1-4612-1656-8 |
| [24] | Duffin, R. J.; Schaeffer, A. C., A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72, 341-366 (1952) · Zbl 0049.32401 |
| [25] | E., Fricain; Mashreghi, J., Theory of de Branges-Rovnyak spaces · Zbl 1159.46307 |
| [26] | Găvruţa, Pasc, On the Feichtinger conjecture, Electron. J. Linear Algebra, 26, 546-552 (2013) · Zbl 1290.46015 · doi:10.13001/1081-3810.1669 |
| [27] | Gregson, Kevin, Extensions of pure states of \({C}^*-\,\) algebras (1986) |
| [28] | Gröchenig, Karlheinz, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math., 18, 2-4, 149-157 (2003) · Zbl 1019.42019 · doi:10.1023/A:1021368609918 |
| [29] | Halpern, Herbert; Kaftal, Victor; Weiss, Gary, The relative Dixmier property in discrete crossed products, J. Funct. Anal., 69, 1, 121-140 (1986) · Zbl 0612.46058 · doi:10.1016/0022-1236(86)90110-2 |
| [30] | Harvey, N. J. A., An introduction to the Kadison-Singer problem and the paving conjecture (2013) |
| [31] | Jaming, P., Cours : analyse fonctionnelle et harmonique (1992-1993), 94, Inversibilité restreinte, problème d’extension de Kadison-Singer et applications à l’analyse harmonique (d’après J. Bourgain et L. Tzafriri), 71-154 (1994), Univ. Paris XI, Orsay · Zbl 0803.47004 |
| [32] | Kadison, Richard V.; Singer, I. M., Extensions of pure states, Amer. J. Math., 81, 383-400 (1959) · Zbl 0086.09704 |
| [33] | Kechris, Alexander S., Classical descriptive set theory, 156, xviii+402 p. pp. (1995), Springer-Verlag, New York · Zbl 0819.04002 · doi:10.1007/978-1-4612-4190-4 |
| [34] | Lata, Sneh; Paulsen, Vern, The Feichtinger conjecture and reproducing kernel Hilbert spaces, Indiana Univ. Math. J., 60, 4, 1303-1317 (2011) · Zbl 1261.46009 · doi:10.1512/iumj.2011.60.4358 |
| [35] | Marcus, Adam; Spielman, Daniel A.; Srivastava, Nikhil, Interlacing families I : bipartite Ramanujan graphs of all degrees (2013) · Zbl 1316.05066 |
| [36] | Marcus, Adam; Spielman, Daniel A.; Srivastava, Nikhil, Interlacing families II : mixed characteristic polynomials and the Kadison-Singer problem (2013) · Zbl 1332.46056 |
| [37] | Marcus, Adam; Spielman, Daniel A.; Srivastava, Nikhil, Ramanujan graphs and the solution to the Kadison-Singer problem (2014) |
| [38] | McKenna, P. J., Discrete Carleson measures and some interpolation problems, Michigan Math. J., 24, 3, 311-319 (1977) · Zbl 0391.30023 |
| [39] | Nikolʼskiĭ, N. K., Treatise on the shift operator, 273, xii+491 p. pp. (1986), Springer-Verlag, Berlin · Zbl 0587.47036 · doi:10.1007/978-3-642-70151-1 |
| [40] | Paulsen, Vern I., A dynamical systems approach to the Kadison-Singer problem, J. Funct. Anal., 255, 1, 120-132 (2008) · Zbl 1156.46041 · doi:10.1016/j.jfa.2008.04.006 |
| [41] | Popa, Sorin, A \({\rm II}_1\) factor approach to the Kadison-Singer problem, Comm. Math. Phys., 332, 1, 379-414 (2014) · Zbl 1306.46060 · doi:10.1007/s00220-014-2055-4 |
| [42] | Reid, G. A., On the Calkin representations, Proc. London Math. Soc. (3), 23, 547-564 (1971) · Zbl 0239.46063 |
| [43] | Seip, Kristian, Interpolation and sampling in spaces of analytic functions, 33, xii+139 p. pp. (2004), American Mathematical Society, Providence, RI · Zbl 1057.30036 · doi:10.1090/ulect/033 |
| [44] | Serre, D., Les matrices : théorie et pratique., vii + 168 p. pp. (2001), Dunod · Zbl 1008.15002 |
| [45] | Shapiro, H. S.; Shields, A. L., On some interpolation problems for analytic functions, Amer. J. Math., 83, 513-532 (1961) · Zbl 0112.29701 |
| [46] | Spielman, Daniel A.; Srivastava, Nikhil, An elementary proof of the restricted invertibility theorem, Israel J. Math., 190, 83-91 (2012) · Zbl 1261.46007 · doi:10.1007/s11856-011-0194-2 |
| [47] | Tanbay, Betül, Pure state extensions and compressibility of the \(l_1\)-algebra, Proc. Amer. Math. Soc., 113, 3, 707-713 (1991) · Zbl 0756.46026 · doi:10.2307/2048605 |
| [48] | Tao, Terry, Real stable polynomials and the Kadison-Singer problem (2013) |
| [49] | Timotin, Dan, The solution to the Kadison-Singer Problem - yet another presentation (2015) |
| [50] | Tropp, Joel A., The random paving property for uniformly bounded matrices, Studia Math., 185, 1, 67-82 (2008) · Zbl 1152.46007 · doi:10.4064/sm185-1-4 |
| [51] | Valette, Alain, Le problème de Kadison-Singer (d’après A. Marcus, D. Spielman et N. Srivastava), Astérisque, 367-368, 451-476 (2015) · Zbl 1356.46002 |
| [52] | Weaver, Nik, A counterexample to a conjecture of Akemann and Anderson, Bull. London Math. Soc., 35, 1, 65-71 (2003) · Zbl 1031.46062 · doi:10.1112/S0024609302001340 |
| [53] | Weaver, Nik, The Kadison-Singer problem in discrepancy theory, Discrete Math., 278, 1-3, 227-239 (2004) · Zbl 1040.46040 · doi:10.1016/S0012-365X(03)00253-X |
| [54] | Youssef, Pierre, Restricted invertibility and the Banach-Mazur distance to the cube, Mathematika, 60, 1, 201-218 (2014) · Zbl 1304.15019 · doi:10.1112/S0025579313000144 |
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