Invariant subspaces for operators in a general II1-factor

Uffe Haagerup; Hanne Schultz

Publications Mathématiques de l'IHÉS (2009)

  • Volume: 109, page 19-111
  • ISSN: 0073-8301

Abstract

top
Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace 𝒦 = 𝒦 T ( B ) affiliated with ℳ, such that the Brown measure of T | 𝒦 is concentrated on B and the Brown measure of P 𝒦 ⊥ T | 𝒦 ⊥ is concentrated on ℂ∖B. Moreover, 𝒦 is T-hyperinvariant and the trace of P 𝒦 is equal to μ T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit A : = lim n → ∞ [ ( T n ) * T n ] 1 2 n exists in the strong operator topology, and the projection onto 𝒦 T ( B ( 0 , r ) ¯ ) is equal to 1[0,r](A), for every rgt;0.

How to cite

top

Haagerup, Uffe, and Schultz, Hanne. "Invariant subspaces for operators in a general II1-factor." Publications Mathématiques de l'IHÉS 109 (2009): 19-111. <http://eudml.org/doc/274348>.

@article{Haagerup2009,
abstract = {Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $\{\mathcal \{K\}\}=\{\mathcal \{K\}\}_\{\mathrm \{T\}\}(B)$ affiliated with ℳ, such that the Brown measure of $\mathrm \{T\}|_\{\{\mathcal \{K\}\}\}$ is concentrated on B and the Brown measure of $\mathrm \{P\}_\{\{\mathcal \{K\}\}^\{\bot \}\}\mathrm \{T\}|_\{\{\mathcal \{K\}\}^\{\bot \}\}$ is concentrated on ℂ∖B. Moreover, $\{\mathcal \{K\}\}$ is T-hyperinvariant and the trace of $\mathrm \{P\}_\{\mathcal \{K\}\}$ is equal to μ T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit $A:=\lim _\{n\rightarrow \infty \}[(\mathrm \{T\}^\{n\})^\{*\}\mathrm \{T\}^\{n\}]^\{\frac\{1\}\{2n\}\}$ exists in the strong operator topology, and the projection onto $\{\mathcal \{K\}\}_\{\mathrm \{T\}\}(\overline\{B(0,r)\})$ is equal to 1[0,r](A), for every rgt;0.},
author = {Haagerup, Uffe, Schultz, Hanne},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {von Neumann factors; Brown measures; invariant subspace; T-hyperinvariant subspace},
language = {eng},
pages = {19-111},
publisher = {Springer-Verlag},
title = {Invariant subspaces for operators in a general II1-factor},
url = {http://eudml.org/doc/274348},
volume = {109},
year = {2009},
}

TY - JOUR
AU - Haagerup, Uffe
AU - Schultz, Hanne
TI - Invariant subspaces for operators in a general II1-factor
JO - Publications Mathématiques de l'IHÉS
PY - 2009
PB - Springer-Verlag
VL - 109
SP - 19
EP - 111
AB - Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace ${\mathcal {K}}={\mathcal {K}}_{\mathrm {T}}(B)$ affiliated with ℳ, such that the Brown measure of $\mathrm {T}|_{{\mathcal {K}}}$ is concentrated on B and the Brown measure of $\mathrm {P}_{{\mathcal {K}}^{\bot }}\mathrm {T}|_{{\mathcal {K}}^{\bot }}$ is concentrated on ℂ∖B. Moreover, ${\mathcal {K}}$ is T-hyperinvariant and the trace of $\mathrm {P}_{\mathcal {K}}$ is equal to μ T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit $A:=\lim _{n\rightarrow \infty }[(\mathrm {T}^{n})^{*}\mathrm {T}^{n}]^{\frac{1}{2n}}$ exists in the strong operator topology, and the projection onto ${\mathcal {K}}_{\mathrm {T}}(\overline{B(0,r)})$ is equal to 1[0,r](A), for every rgt;0.
LA - eng
KW - von Neumann factors; Brown measures; invariant subspace; T-hyperinvariant subspace
UR - http://eudml.org/doc/274348
ER -

References

top
  1. BL. P. Biane, F. Lehner, Computation of some examples of Brown’s spectral measure in free probability, Colloq. Math.90 (2001), p. 181-211 Zbl0988.22004MR1876844
  2. Br. L. G. Brown, Lidskii’s theorem in the type II case, in: Geometric Methods in Operator Algebras (Kyoto 1983), Pitman Res. Notes Math. Ser. 123 (1986), Longman Sci. Tech., Harlow Zbl0646.46058MR866489
  3. Co. A. Connes, Classification of injective factors, Ann. Math.104 (1976), p. 73-115 Zbl0343.46042MR454659
  4. D. K. Dykema, Hyperinvariant subspaces for some B-circular operators, Math. Ann.333 (2005), p. 485-523 Zbl1083.47010MR2198797
  5. DH1. K. Dykema, U. Haagerup, Invariant subspaces of Voiculescu’s circular operator, Geom. Funct. Anal.11 (2001), p. 693-741 Zbl1023.47005MR1866799
  6. DH2. K. Dykema, U. Haagerup, DT-operators and decomposability of Voiculescu’s circular operator, Am. J. Math.126 (2004), p. 121-189 Zbl1054.47026MR2033566
  7. DH3. K. Dykema, U. Haagerup, Invariant subspaces of the quasinilpotent DT-operator, J. Funct. Anal.209 (2004), p. 332-366 Zbl1087.47006MR2044226
  8. FK. T. Fack, H. Kosaki, Generalized s-numbers of τ-measurable operators, Pac. J. Math.123 (1986), p. 269-300 Zbl0617.46063MR840845
  9. FuKa. B. Fuglede, R. V. Kadison, Determinant theory in finite factors, Ann. Math.55 (1952), p. 520-530 Zbl0046.33604MR52696
  10. Fo. G. B. Folland, Real Analysis, Modern Techniques and their Applications, (1984), Wiley, New York Zbl0924.28001MR767633
  11. H. U. Haagerup, Spectral decomposition of all operators in a II1-factor which is embeddable in R ω , Unpublished lecture notes, MSRI, 2001. 
  12. H2. U. Haagerup, Random matrices, free probability and the invariant subspace problem relative to a von Neumann algebra, in Proceedings of the International Congress of Mathematics, vol. 1, pp. 273–290, 2002. Zbl1047.46043MR1989189
  13. HL. U. Haagerup, F. Larsen, Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras, J. Funct. Anal.176 (2000), p. 331-367 Zbl0984.46042MR1784419
  14. HS. U. Haagerup, H. Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra, Math. Scand.100 (2007), p. 209-263 Zbl1168.46039MR2339369
  15. HT. U. Haagerup, S. Thorbjørnsen, A new application of random matrices: Ext(C red * (F 2)) is not a group, Ann. Math.162 (2005), p. 711-775 Zbl1103.46032MR2183281
  16. HW. U. Haagerup, C. Winsløw, The Effros-Maréchal topology in the space of von Neumann algebras, II, J. Funct. Anal.171 (2000), p. 401-431 Zbl0982.46045MR1745629
  17. KR. R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, (1983), Academic Press, New York Zbl0518.46046MR719020
  18. KR2. R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II, (1986), Academic Press, Orlando Zbl0601.46054MR859186
  19. Ka. N. J. Kalton, Analytic functions in non-locally convex spaces and applications, Stud. Math.83 (1986), p. 275-303 Zbl0634.46038MR850829
  20. La. S. Lang, Real and Functional Analysis, Graduate Texts in Mathematics 142 (1993), Springer, New York Zbl0831.46001MR1216137
  21. LN. K. B. Laursen, M. M. Neumann, An Introduction to Local Spectral Theory, (2000), Clarendon, New York Zbl0957.47004MR1747914
  22. Ru. W. Rudin, Real and Complex Analysis, (1987), McGraw-Hill, New York Zbl0925.00005MR924157
  23. Sh. D. Shlyakhtenko, Some applications of freeness with amalgamation, J. Reine Angew. Math.500 (1998), p. 191-212 Zbl0926.46046MR1637501
  24. SS. P. Sniady, R. Speicher, Continuous families of invariant subspaces for R-diagonal operators, Invent. Math.146 (2001), p. 329-363 Zbl1032.46077MR1865398
  25. TuWa. R. Turpin, L. Waelbroeck, Intégration et fonctions holomorphes dans les espaces localement pseudo-convexes, C.R. Acad. Sci. Paris Sér. A-B267 (1968), p. 160-162 Zbl0159.42603MR234281
  26. V1. D. Voiculescu, Circular and semicircular systems and free product factors, in: Operator Algebras, Unitary Representations, Algebras, and Invariant Theory (Paris, 1989), Progr. Math. 92 (1990), Birkhäuser, Boston Zbl0744.46055MR1103585
  27. VDN. D. Voiculescu, K. Dykema, A. Nica, Free Random Variables, CRM Monograph Series 1 (1992), American Mathematical Society, Providence Zbl0795.46049MR1217253
  28. Wa. L. Waelbroeck, Topological Vector Spaces and Algebras, Lecture Notes in Mathematics 230 (1971), Springer, Berlin Zbl0225.46001MR467234

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.