Brown measures of sets of commuting operators in a type II1 factor

Using the spectral subspaces obtained in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II1-factor, preprint, 2005], Brown’s results (cf. [L.G. Brown, Lidskii’s theorem in the type II case, in: H. Araki, E. Effros (Eds.), Geometric Methods in Operator Algebras, Kyoto, 1983, in: Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., 1986, pp. 1–35]) on the Brown measure of an operator in a type II1 factor (M, τ) are generalized to finite sets of commuting operators in M. It is shown that whenever T1, . . . , Tn ∈M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure μT1,...,Tn on B(Cn) such that for all α1, . . . , αn ∈ C, τ log |α1T1 +···+αnTn − 1| = Cn log |α1z1 +···+αnzn −1|dμT1,...,Tn (z1, . . . , zn). Moreover, for every polynomial q in n commuting variables, μq(T1,...,Tn) is the push-forward measure of μT1,...,Tn via the map q :Cn→C. In addition it is shown that, as in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II1-factor, preprint, 2005], for every Borel set B ⊆ Cn there is a maximal closed T1-, . . . , Tn-invariant subspace K affiliated withM, such that μT1|K,...,Tn|K is concentrated on B.Moreover, τ(PK) = μT1,...,Tn(B). This generalizes the main result from [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II1-factor, preprint, 2005] to n-tuples of commuting operators inM. © 2006 Elsevier Inc. All rights reserved.