Invariant subspaces for operators in a general II1-factor

Haagerup, Uffe ; Schultz, Hanne

Publications Mathématiques de l'IHÉS, Tome 110 (2009), p. 19-111 / Harvested from Numdam

Résumé

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 r>0.

Détail

Publié le : 2009-01-01

MR 2511586 | Zbl 1178.46058

DOI : https://doi.org/10.1007/s10240-009-0018-7

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