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Dykema, Kenneth J.; Fang, Junsheng; Hadwin, Donald W.; Smith, Roger R.

The carpenter and Schur-Horn problems for masas in finite factors. (English) Zbl 1292.46040

Ill. J. Math. 56, No. 4, 1313-1329 (2012).
Summary: Two classical theorems in matrix theory, due to I. Schur [Sitzungsber. Berl. Math. Ges. 22, 9–20 (1923; JFM 49.0054.01)] and A. Horn [Am. J. Math. 76, 620–630 (1954; Zbl 0055.24601)], relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur-Horn problem, where matrix algebras and diagonals are replaced respectively, by finite factors and maximal Abelian self-adjoint subalgebras (masas) [W. Arveson and R. V. Kadison, Contemp. Math. 414, 247–263 (2006; Zbl 1113.46064)]. There is a special case of the problem, called the carpenter problem [R. V. Kadison, Proc. Natl. Acad. Sci. USA 99, No. 7, 4178–4184 (2002; Zbl 1013.46049); ibid., No. 8, 5217–5222 (2002; Zbl 1013.46050)], which can be stated as follows: for a masa \(A\) in a finite factor \(M\) with conditional expectation \(\mathbb{E}_{A}\), can each \(x \in A\) with \(0\leq x \leq1\) be expressed as \(\mathbb{E}_{A}(p)\) for a projection \(p \in M\)? { } In this paper, we investigate these problems for various masas. We give positive solutions for the generator and radial masas in free group factors, and we also solve affirmatively a weaker form of the Schur-Horn problem for the Cartan masa in the hyperfinite factor.

Cited in 11 Documents

MSC:

46L10 General theory of von Neumann algebras
15A42 Inequalities involving eigenvalues and eigenvectors

Keywords:

Schur-Horn problem; finite factor; maximal Abelian self-adjoint subalgebras; carpenter problem; masa; hyperfinite factor

Citations:

JFM 49.0054.01; Zbl 0055.24601; Zbl 1113.46064; Zbl 1013.46049; Zbl 1013.46050
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Full Text: arXiv Euclid

References:

[1] \beginbarticle \bauthor\binitsM. \bsnmArgerami and \bauthor\binitsP. \bsnmMassey, \batitleA Schur-Horn theorem in \(\mathrm{II}_1\) factors, \bjtitleIndiana Univ. Math. J. \bvolume56 (\byear2007), page 2051-\blpage2059. \endbarticle \endbibitem · Zbl 1136.46043 · doi:10.1512/iumj.2007.56.3113
[2] \beginbarticle \bauthor\binitsM. \bsnmArgerami and \bauthor\binitsP. \bsnmMassey, \batitleTowards the carpenter’s theorem, \bjtitleProc. Amer. Math. Soc. \bvolume137 (\byear2009), page 3679-\blpage3687. \endbarticle \endbibitem · Zbl 1183.46058 · doi:10.1090/S0002-9939-09-09999-7
[3] \beginbchapter \bauthor\binitsW. \bsnmArveson and \bauthor\binitsR. V. \bsnmKadison, \bctitleDiagonals of self-adjoint operators, \bbtitleOperator theory, operator algebras, and applications, \bsertitleContemp. Math., vol. \bseriesno414, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear2006, pp. page 247-\blpage263. \endbchapter \endbibitem · doi:10.1090/conm/414/07814
[4] \beginbarticle \bauthor\binitsH. \bsnmBercovici and \bauthor\binitsW. S. \bsnmLi, \batitleEigenvalue inequalities in an embeddable factor, \bjtitleProc. Amer. Math. Soc. \bvolume134 (\byear2006), page 75-\blpage80. \endbarticle \endbibitem · Zbl 1086.15019 · doi:10.1090/S0002-9939-05-07952-9
[5] \beginbarticle \bauthor\binitsA. \bsnmConnes, \bauthor\binitsJ. \bsnmFeldman and \bauthor\binitsB. \bsnmWeiss, \batitleAn amenable equivalence relation is generated by a single transformation, \bjtitleErgodic Theory Dynam. Systems \bvolume1 (\byear1981), page 431-\blpage450. \endbarticle \endbibitem · Zbl 0491.28018 · doi:10.1017/S014338570000136X
[6] \beginbarticle \bauthor\binitsT. \bsnmFack and \bauthor\binitsH. \bsnmKosaki, \batitleGeneralized \(s\)-numbers of \(\tau\)-measurable operators, \bjtitlePacific J. Math. \bvolume123 (\byear1986), page 269-\blpage300. \endbarticle \endbibitem · Zbl 0617.46063 · doi:10.2140/pjm.1986.123.269
[7] \beginbarticle \bauthor\binitsF. \bsnmHiai, \batitleMajorization and stochastic maps in von Neumann algebras, \bjtitleJ. Math. Anal. Appl. \bvolume127 (\byear1987), page 18-\blpage48. \endbarticle \endbibitem · Zbl 0634.46051 · doi:10.1016/0022-247X(87)90138-7
[8] \beginbchapter \bauthor\binitsF. \bsnmHiai, \bctitleSpectral majorization between normal operators in von Neumann algebras, \bbtitleOperator algebras and operator theory (Craiova, 1989), \bsertitlePitman Res. Notes Math. Ser., vol. \bseriesno271, \bpublisherLongman, \blocationHarlow, \byear1992, pp. page 78-\blpage115. \endbchapter \endbibitem
[9] \beginbarticle \bauthor\binitsA. \bsnmHorn, \batitleDoubly stochastic matrices and the diagonal of a rotation matrix, \bjtitleAmer. J. Math. \bvolume76 (\byear1954), page 620-\blpage630. \endbarticle \endbibitem · Zbl 0055.24601 · doi:10.2307/2372705
[10] \beginbarticle \bauthor\binitsR. V. \bsnmKadison, \batitleThe Pythagorean theorem. I. The finite case, \bjtitleProc. Natl. Acad. Sci. USA \bvolume99 (\byear2002), page 4178-\blpage4184. \endbarticle \endbibitem · Zbl 1013.46049 · doi:10.1073/pnas.032677199
[11] \beginbarticle \bauthor\binitsR. V. \bsnmKadison, \batitleThe Pythagorean theorem. II. The infinite discrete case, \bjtitleProc. Natl. Acad. Sci. USA \bvolume99 (\byear2002), page 5217-\blpage5222. \endbarticle \endbibitem · Zbl 1013.46050 · doi:10.1073/pnas.032677299
[12] \beginbarticle \bauthor\binitsE. \bsnmKamei, \batitleMajorization in finite factors, \bjtitleMath. Japon. \bvolume28 (\byear1983), page 495-\blpage499. \endbarticle \endbibitem
[13] \beginbarticle \bauthor\binitsF. J. \bsnmMurray and \bauthor\binitsJ. \bsnmvon Neumann, \batitleOn rings of operators, \bjtitleAnn. of Math. (2) \bvolume37 (\byear1936), page 116-\blpage229. \endbarticle \endbibitem · Zbl 0014.16101 · doi:10.2307/1968693
[14] \beginbarticle \bauthor\binitsD. \bsnmPetz, \batitleSpectral scale of selfadjoint operators and trace inequalities, \bjtitleJ. Math. Anal. Appl. \bvolume109 (\byear1985), page 74-\blpage82. \endbarticle \endbibitem · Zbl 0655.47032 · doi:10.1016/0022-247X(85)90176-3
[15] \beginbarticle \bauthor\binitsS. \bsnmPopa, \batitleOrthogonal pairs of *-subalgebras in finite von Neumann algebras, \bjtitleJ. Operator Theory \bvolume9 (\byear1983), page 253-\blpage268. \endbarticle \endbibitem
[16] \beginbarticle \bauthor\binitsS. \bsnmPopa, \batitleNotes on Cartan subalgebras in type \(\mathrm{II}_1\) factors, \bjtitleMath. Scand. \bvolume57 (\byear1985), page 171-\blpage188. \endbarticle \endbibitem
[17] \beginbarticle \bauthor\binitsI. \bsnmSchur, eine Klasse von Mittlebildungen mit Anwendungen auf der Determinantentheorie, \bjtitleSitzungsber. Berliner Mat. Ges. \bvolume22 (\byear1923), page 9-\blpage29. \endbarticle \endbibitem
[18] \beginbbook \bauthor\binitsA. M. \bsnmSinclair and \bauthor\binitsR. R. \bsnmSmith, \bbtitleFinite von Neumann algebras and masas, \bsertitleLondon Mathematical Society Lecture Note Series, vol. \bseriesno351, \bpublisherCambridge University Press, \blocationCambridge, \byear2008. \endbbook \endbibitem · doi:10.1017/CBO9780511666230
[19] \beginbbook \bauthor\binitsD. \bsnmVoiculescu, \bbtitleSymmetries of some reduced free product \(C^{\ast}\)-algebras, \bsertitleOperator algebras, unitary representations, enveloping algebras, and invariant theory, Lecture Notes in Mathematics, vol. \bseriesno1132, \bpublisherSpringer, \blocationBerlin, \byear1985, pp. page 556-\blpage588. \endbbook \endbibitem · doi:10.1007/BFb0074909
[20] \beginbchapter \bauthor\binitsD. \bsnmVoiculescu, \bauthor\binitsK. \bsnmDykema and \bauthor\binitsA. \bsnmNica, \bctitleFree random variables, \bbtitleCRM Monograph Series, vol. 1, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1992. \endbchapter \endbibitem
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