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Brown measure

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Probability measure on a complex plane

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Inmathematics, theBrown measure of an operator in a finitefactor is aprobability measure on the complex plane which may be viewed as an analog of thespectral counting measure (based onalgebraic multiplicity) of matrices.

It is named afterLawrence G. Brown.

Definition

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Let ${\mathcal {M}}$ be a finite factor with the canonical normalized trace $\tau$ and let $I {\displaystyle I}$ 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 asubharmonic function and itsLaplacian in thedistributional 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 $A . {\displaystyle A.}$ Here the Laplace operator $\nabla ^{2}$ is complex.

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

References

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

Measure theory

Basic concepts

Sets

Types ofmeasures

Particular measures

Maps

Main results

Carathéodory's extension theorem
Convergence theorems
Decomposition theorems
Egorov's
Fatou's lemma
Fubini's
- Fubini–Tonelli
Hölder's inequality
Minkowski inequality
Radon–Nikodym
Riesz–Markov–Kakutani representation theorem

Other results

Disintegration theorem Lifting theory Lebesgue's density theorem Lebesgue differentiation theorem Sard's theorem Vitali–Hahn–Saks theorem
ForLebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

v t e Spectral theory and^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With anApproximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

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