II_1型因子中交换算子集的Brown测度
Brown measures of sets of commuting operators in a type II_1 factor
摘要：
利用[HS]中得到的谱子空间，将Brown关于II_1型因子（M，tr）中算子的Brown测度的结果推广到M中交换算子的有限集.结果表明，只要T_1，. T_n在M中是相互交换的算子，在C^n上存在一个且只有一个紧支撑的Borel概率测度mu_{T_1,..., T_n}，使得对所有α_1，.，alpha_n in C，tr（log| α_1 T_1+ ... + α_n T_n - 1|）是log的积分|alpha_1 z_1 +. + α_n z_n-1| w.r.t mu_{T_1,...,T_n}。此外，对于n个交换变量中的每个多项式q，mu_{q(T_1,..., T_n)}是mu_{T_1,...,T_n}通过映射q的推进测度。此外，如[HS]所示，对C^n中的每个Borelset B，存在一个极大闭T_1-，.，T_n-隶属于M的不变子空间K，使得mu_{T_1|_K,..., T_n|_K}集中于B.此外，tr（P_K）= mu_{T_1,...,T_n}（B）。这将[HS]的主要结果推广到M中交换算子的n元组。
英文摘要：
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.
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