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

Using the spectral subspaces obtained in [HS], Brown’s results (cf. [Bro]) 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(C) 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 : C → C. In addition it is shown that, as in [HS], for every Borelset B ⊆ C there is a maximal closed T1-,..., Tn-invariant subspace K affiliated with M, such that μT1|K,...,Tn|K is concentrated on B.