Brown measure

In mathematics, the Brown measure of an operator in a finite factor is a probability measure on the complex plane which may be viewed as an analog of the spectral counting measure (based on algebraic multiplicity) of matrices.

It is named after Lawrence G. Brown.

Definition

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

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

References

Brown, Lawrence (1986), "Lidskii's theorem in the type $II$ 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 $II_{1}$ factor", Publ. Math. Inst. Hautes Études Sci., 109: 19–111, arXiv:math/0611256, doi:10.1007/s10240-009-0018-7, S2CID 11359935.