Brown measure

Let ${\displaystyle {\mathcal {M}}}$ be a finite factor with the canonical normalized trace ${\displaystyle \tau }$ and let ${\displaystyle I}$ be the identity operator. For every operator ${\displaystyle A\in {\mathcal {M}},}$ the function $${\displaystyle \lambda \mapsto \tau (\log \left|A-\lambda I\right|),\;\lambda \in \mathbb {C} ,}$$ is a subharmonic function and its Laplacian in the distributional sense is a probability measure on ${\displaystyle \mathbb {C} }$ $${\displaystyle \mu _{A}(\mathrm {d} (a+b\mathbb {i} )):={\frac {1}{2\pi }}\nabla ^{2}\tau (\log \left|A-(a+b\mathbb {i} )I\right|)\mathrm {d} a\mathrm {d} b}$$ which is called the Brown measure of ${\displaystyle A.}$ Here the Laplace operator ${\displaystyle \nabla ^{2}}$ is complex.

The subharmonic function can also be written in terms of the Fuglede−Kadison determinant ${\displaystyle \Delta _{FK}}$ as follows $${\displaystyle \lambda \mapsto \log \Delta _{FK}(A-\lambda I),\;\lambda \in \mathbb {C} .}$$

• Direct integral – Generalization of the concept of direct sum in mathematics

• Brown, Lawrence (1986), "Lidskii's theorem in the type ${\displaystyle II}$ case", Pitman Res. Notes Math. Ser., 123, Longman Sci. Tech., Harlow: 1–35. Geometric methods in operator algebras (Kyoto, 1983).

• Haagerup, Uffe; Schultz, Hanne (2009), "Brown measures of unbounded operators in a general ${\displaystyle II_{1}}$ factor", Publ. Math. Inst. Hautes Études Sci., 109: 19–111, arXiv:math/0611256, doi:10.1007/s10240-009-0018-7, S2CID 11359935.