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)(M,τ) are generalized to finite sets of commuting operators in MM. It is shown that whenever T1,…,Tn∈MT1,…,Tn∈M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure μT1,…,TnμT1,…,Tn on B(Cn)B(Cn) such that for all α1,…,αn∈Cα1,…,αn∈C,τ(log|α1T1+⋯+αnTn−1|)=∫Cnlog|α1z1+⋯+αnzn−1|dμT1,…,Tn(z1,…,zn). Moreover, for every polynomial q in n   commuting variables, μq(T1,…,Tn)μq(T1,…,Tn) is the push-forward measure of μT1,…,TnμT1,…,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⊆CnB⊆Cn there is a maximal closed T1-,…,TnT1-,…,Tn-invariant subspace KK affiliated with MM, such that μT1|K,…,Tn|KμT1|K,…,Tn|K is concentrated on B  . Moreover, τ(PK)=μT1,…,Tn(B)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 in MM.