Brown measures of sets of commuting operators in a type II_1 factor

Abstract

Using the spectral subspaces obtained in [HS], Brown's results on the Brown measure of an operator in a type II_1 factor (M,tr) are generalized to finite sets of commuting operators in M. It is shown that whenever T_1,..., T_n in M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure mu_{T_1,..., T_n} on C^n such that for all alpha_1,..., alpha_n in C, tr(log|alpha_1 T_1+ ... + alpha_n T_n - 1|) is the integral of log|alpha_1 z_1 + ... + alpha_n z_n-1| w.r.t mu_{T_1,...,T_n}. Moreover, for every polynomial q in n commuting variables, mu_{q(T_1,..., T_n)} is the push-forward measure of mu_{T_1,...,T_n} via the map q. In addition it is shown that, as in [HS], for every Borelset B in C^n there is a maximal closed T_1-,..., T_n-invariant subspace K affiliated with M, such that mu_{T_1|_K,..., T_n|_K} is concentrated on B. Moreover, tr(P_K)=mu_{T_1,...,T_n}(B). This generalizes the main result from [HS] to n-tuples of commuting operators in M.