Brown measure explained

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

l{M} be a finite factor with the canonical normalized trace \tau and let I be the identity operator. For every operator A \in l{M}, the function$$\backslash lambda\; \backslash mapsto\; \backslash tau(\backslash log\; \backslash left|A-\backslash lambda\; I\backslash right|),\; \backslash ;\; \backslash lambda\; \backslash in\; \backslash Complex,$$is a subharmonic function and its Laplacian in the distributional sense is a probability measure on \Complex $$\backslash mu\_A(\backslash mathrm(a+b\backslash mathbb))\; :=\; \backslash frac\backslash nabla^2\; \backslash tau(\backslash log\; \backslash left|A-(a+b\backslash mathbb)\; I\backslash right|)\backslash mathrma\backslash mathrmb$$which is called the Brown measure of A. Here the Laplace operator \nabla ^{2} is complex. \Delta _{FK} as follows$$\backslash lambda\; \backslash mapsto\; \backslash log\backslash Delta\_(A-\backslash lambda\; I),\; \backslash ;\; \backslash lambda\; \backslash in\; \backslash Complex.$$

References

• . Geometric methods in operator algebras (Kyoto, 1983).
• .