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

     author = {Haagerup, Uffe and Schultz, Hanne},
     title = {Invariant subspaces for operators in a general {II1-factor}},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {19--111},
     publisher = {Springer-Verlag},
     volume = {109},
     year = {2009},
     doi = {10.1007/s10240-009-0018-7},
     zbl = {1178.46058},
     mrnumber = {2511586},
     language = {en},
     url = {}
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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.

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