Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T05:29:01.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Ranges in II1 Factors

Part of: Basic linear algebra Selfadjoint operator algebras General theory of linear operators Individual linear operators as elements of algebraic systems

Published online by Cambridge University Press:  16 March 2017

Ken Dykema  and
Paul Skoufranis
Ken Dykema*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA (kdykema@math.tamu.edu; pskoufra@math.tamu.edu)
Paul Skoufranis*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA (kdykema@math.tamu.edu; pskoufra@math.tamu.edu)
*
*Corresponding author.
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we generalize the notion of the C-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator C, the C-numerical range of such an operator is defined; it is a compact, convex subset of ℂ. We explicitly describe the C-numerical ranges of several operators and classes of operators.

Keywords

II1 FactorsNumerical RangeGeneralized Numerical Range

MSC classification

Primary: 46L10: General theory of von Neumann algebras 47C15: Operators in $C^*$- or von Neumann algebras 47A12: Numerical range, numerical radius 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 31 - 55
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Akemann, C. and Anderson, J., A geometrical spectral theory for n-tuples of self-adjoint operators in finite von Neumann algebras: II, Pacific J. Math. 205 (2002), 257–285.CrossRefGoogle Scholar
2. Akemann, C. and Anderson, J., The spectral scale and the k-numerical range, Glasgow Math. J. 45 (2003), 225–238.CrossRefGoogle Scholar
3. Akemann, C. and Anderson, J., The spectral scale and the numerical range, Internat. J. Math. 14 (2003), 171–189.CrossRefGoogle Scholar
4. Akemann, C., Anderson, J. and Weaver, N., A geometrical spectral theory for n-tuples of self-adjoint operators in finite von Neumann algebras, J. Funct. Anal. 165 (1999), 258–292.CrossRefGoogle Scholar
5. Ando, T., Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Alg. Applc. 118 (1989), 163–248.CrossRefGoogle Scholar
6. Argerami, M. and Massey, P., A Schur–Horn theorem in II1 factors, Indiana Univ. Math. J. 56 (2007), 2051–2060.CrossRefGoogle Scholar
7. Argerami, M. and Massey, P., The local form of doubly stochastic maps and joint majorization in II1 factors, Integral Equations Operator Theory 61(1) (2008), 1–19.CrossRefGoogle Scholar
8. Dykema, K., Fang, J., Hadwin, D. and Smith, R., The carpenter and Schur–Horn problems for MASAs in finite factors, Illinois J. Math. 56 (2012), 1313–1329.CrossRefGoogle Scholar
9. Dykema, K. and Haagerup, U., DT-operators and decomposability of Voiculescu's circular operator, Amer. J. Math. 126 (2004), 121–189.CrossRefGoogle Scholar
10. Dykema, K. and Haagerup, U., Invariant subspaces of the quasinilpotent DT-operator, J. Funct. Anal. 209 (2004), 332–366.CrossRefGoogle Scholar
11. Fack, T., Sur la notion de valuer caractéristique, J. Operator Theory 7 (1982), 207–333.Google Scholar
12. Fack, T. and Kosaki, H., Generalized s-numbers of τ-measurable operators, Pacific J. Math 123 (1986), 269–300.CrossRefGoogle Scholar
13. Fillmore, P. A. and Williams, J. P., Some convexity theorems for matrices, Glasgow Math J. 11(2) (1971), 110–117.CrossRefGoogle Scholar
14. Goldberg, M. and Straus, E., Inclusion relations involving k-numerical ranges, Linear Algebra Appl. 15(3) (1976), 261–270.CrossRefGoogle Scholar
15. Goldberg, M. and Straus, E., Elementary inclusion relations for generalized numerical ranges, Linear Algebra Appl. 18(1) (1977), 1–24.CrossRefGoogle Scholar
16. Gustafson, K., The Toeplitz-Hausdorff theorem for linear operators, Proc. Amer. Math. Soc. 25 (1970), 203–204.Google Scholar
17. Halmos, P. R., A Hilbert Space Problem Book, 1 (Princeton: van Nostrand, 1967).Google Scholar
18. Hardy, G. H., Littlewood, J. E. and Pólya, G., Some simple inequalities satisfied by convex functions, Messenger Math. 58 (1929) 145–152.Google Scholar
19. Hausdorff, F., Der Wertvorrat einer Bilinearform, Math. Z. 3(1) (1919), 314–316.CrossRefGoogle Scholar
20. Hiai, F., Majorization and stochastic maps in von Neumann algebras, Journal of Mathematical Analysis and Applications. 127 (1987), 18–48.CrossRefGoogle Scholar
21. Hiai, F., Spectral majorization between normal operators in von Neumann algebras, Operator algebras and operator theory, Craiova, 1989, Pitman Res. Notes Math. Ser., 271, Longman Sci. Tech., Harlow, 1992, 78–115.Google Scholar
22. Hiai, F. and Nakamura, Y., Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc. 323 (1991), 1–38.CrossRefGoogle Scholar
23. Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620–630.CrossRefGoogle Scholar
24. Kamei, E., Majorization in finite factors, Math. Japon. 28 (1983), 495–499.Google Scholar
25. Kamei, E., Double stochasticity in finite factors, Math. Japon. 29 (1984), 903–907.Google Scholar
26. Kamei, E., An order on statistical operators implicitly introduced by von Neumann, Math. Japon. 30 (1985), 891–895.Google Scholar
27. Keeler, D., Rodman, L. and Spitkovsky, I., The numerical range of 3×3 matrices, Linear Algebra Appl. 252(1) (1997), 115–139.CrossRefGoogle Scholar
28. Kippenhahn, R., Über den Wertevorrat einer Matrix, Math. Nachr. 6 (1951), 193–228.CrossRefGoogle Scholar
29. Kippenhahn, R., On the numerical range of a matrix, Translated from the German by Paul F. Zachlin and Michiel E. Hochstenbach, Linear and Multilinear Algebra 56 (2008), 185–225.Google Scholar
30. Larsen, F., Brown Measures and R-diagonal Elements in Finite von Neumann Algebras, 1999, Ph.D. Thesis, University of Southern Denmark.Google Scholar
31. Murray, F. J. and von Neumann, J., On rings of operators, Ann. of Math. (2) 37 (1936), 116–229.CrossRefGoogle Scholar
32. Petz, D., Spectral scale of self-adjoint operators and trace inequalities, J. Math. Anal. Appl. 109 (1985), 74–82.CrossRefGoogle Scholar
33. Poon, Y. T., Another proof of a result of Westwick, Linear Algebra Appl. 9 (1980), 35–37.Google Scholar
34. Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berl. Math. Ges. 22 (1923), 9–20.Google Scholar
35. Tucci, G., Some quasinilpotent generators of the hyperfinite II1 factor, J. Funct. Anal. 254 (2008), 2969–2994.CrossRefGoogle Scholar
36. Westwick, R., A theorem on numerical range, Linear Algebra Appl. 2 (1975), 311–315.Google Scholar