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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.


Let be a finite factor with the canonical normalized trace and let be the identity operator. For every operator the function

is a subharmonic function and its Laplacian in the distributional sense is a probability measure on
which is called the Brown measure of Here the Laplace operator is complex.

The subharmonic function can also be written in terms of the Fuglede−Kadison determinant as follows

See also

  • Direct integral – generalization of the concept of direct sum


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