Invariant subspaces for operators in a general II1-factor
Publications Mathématiques de l'IHÉS, Volume 109  (2009), p. 19-111

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 2n exists in the strong operator topology, and the projection onto 𝒦 T (B(0,r) ¯) is equal to 1[0,r](A), for every r>0.

@article{PMIHES_2009__109__19_0,
     author = {Haagerup, Uffe and Schultz, Hanne},
     title = {Invariant subspaces for operators in a general II1-factor},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     publisher = {Springer-Verlag},
     volume = {109},
     year = {2009},
     pages = {19-111},
     doi = {10.1007/s10240-009-0018-7},
     zbl = {1178.46058},
     mrnumber = {2511586},
     language = {en},
     url = {http://www.numdam.org/item/PMIHES_2009__109__19_0}
}
Haagerup, Uffe; Schultz, Hanne. Invariant subspaces for operators in a general II1-factor. Publications Mathématiques de l'IHÉS, Volume 109 (2009) , pp. 19-111. doi : 10.1007/s10240-009-0018-7. http://www.numdam.org/item/PMIHES_2009__109__19_0/

BL. P. Biane, F. Lehner, Computation of some examples of Brown's spectral measure in free probability, Colloq. Math. 90 (2001), p. 181-211 | MR 1876844 | Zbl 0988.22004

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 | MR 866489 | Zbl 0646.46058

Co. A. Connes, Classification of injective factors, Ann. Math. 104 (1976), p. 73-115 | MR 454659 | Zbl 0343.46042

D. K. Dykema, Hyperinvariant subspaces for some B-circular operators, Math. Ann. 333 (2005), p. 485-523 | MR 2198797 | Zbl 1083.47010

DH1. K. Dykema, U. Haagerup, Invariant subspaces of Voiculescu's circular operator, Geom. Funct. Anal. 11 (2001), p. 693-741 | MR 1866799 | Zbl 1023.47005

DH2. K. Dykema, U. Haagerup, DT-operators and decomposability of Voiculescu's circular operator, Am. J. Math. 126 (2004), p. 121-189 | MR 2033566 | Zbl 1054.47026

DH3. K. Dykema, U. Haagerup, Invariant subspaces of the quasinilpotent DT-operator, J. Funct. Anal. 209 (2004), p. 332-366 | MR 2044226 | Zbl 1087.47006

FK. T. Fack, H. Kosaki, Generalized s-numbers of τ-measurable operators, Pac. J. Math. 123 (1986), p. 269-300 | MR 840845 | Zbl 0617.46063

FuKa. B. Fuglede, R. V. Kadison, Determinant theory in finite factors, Ann. Math. 55 (1952), p. 520-530 | MR 52696 | Zbl 0046.33604

Fo. G. B. Folland, Real Analysis, Modern Techniques and their Applications, (1984), Wiley, New York | MR 767633 | Zbl 0549.28001 | Zbl 0924.28001

H. U. Haagerup, Spectral decomposition of all operators in a II1-factor which is embeddable in R ω , Unpublished lecture notes, MSRI, 2001.

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. | MR 1989189 | Zbl 1047.46043

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 | MR 1784419 | Zbl 0984.46042

HS. U. Haagerup, H. Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra, Math. Scand. 100 (2007), p. 209-263 | MR 2339369 | Zbl 1168.46039

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 | MR 2183281 | Zbl 1103.46032

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 | MR 1745629 | Zbl 0982.46045

KR. R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I, (1983), Academic Press, New York | MR 719020 | Zbl 0518.46046

KR2. R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II, (1986), Academic Press, Orlando | MR 859186 | Zbl 0601.46054

Ka. N. J. Kalton, Analytic functions in non-locally convex spaces and applications, Stud. Math. 83 (1986), p. 275-303 | MR 850829 | Zbl 0634.46038

La. S. Lang, Real and Functional Analysis, Graduate Texts in Mathematics 142 (1993), Springer, New York | MR 1216137 | Zbl 0831.46001

LN. K. B. Laursen, M. M. Neumann, An Introduction to Local Spectral Theory, (2000), Clarendon, New York | MR 1747914 | Zbl 0957.47004

Ru. W. Rudin, Real and Complex Analysis, (1987), McGraw-Hill, New York | MR 924157 | Zbl 0925.00005

Sh. D. Shlyakhtenko, Some applications of freeness with amalgamation, J. Reine Angew. Math. 500 (1998), p. 191-212 | MR 1637501 | Zbl 0926.46046

SS. P. Sniady, R. Speicher, Continuous families of invariant subspaces for R-diagonal operators, Invent. Math. 146 (2001), p. 329-363 | MR 1865398 | Zbl 1032.46077

TuWa. R. Turpin, L. Waelbroeck, Intégration et fonctions holomorphes dans les espaces localement pseudo-convexes, C.R. Acad. Sci. Paris Sér. A-B 267 (1968), p. 160-162 | MR 234281 | Zbl 0159.42603

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 | MR 1103585 | Zbl 0744.46055

VDN. D. Voiculescu, K. Dykema, A. Nica, Free Random Variables, CRM Monograph Series 1 (1992), American Mathematical Society, Providence | MR 1217253 | Zbl 0795.46049

Wa. L. Waelbroeck, Topological Vector Spaces and Algebras, Lecture Notes in Mathematics 230 (1971), Springer, Berlin | MR 467234 | Zbl 0225.46001