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.
be a finite factor with the canonical normalized trace
τ
be the identity operator.
λ ↦ τ ( log
− λ
is a subharmonic function and its Laplacian in the distributional sense is a probability measure on
τ ( log
which is called the Brown measure of
Here the Laplace operator
The subharmonic function can also be written in terms of the Fuglede−Kadison determinant
λ ↦ log