Invariant subspaces for operators in a general II₁-factor

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 r>0.