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254
Large deviation for Wigner’s law and Voiculescu’s noncommutative entropy
 PROBAB. THEORY RELATED FIELDS 108
, 1997
"... We study the spectral measure of Gaussian Wigner’s matrices and prove that it satisfies a large deviation principle. We show that the good rate function which governs this principle achieves its minimum value at Wigner’s semicircular law, which entails the convergence of the spectral measure to the ..."
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Cited by 118 (16 self)
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We study the spectral measure of Gaussian Wigner’s matrices and prove that it satisfies a large deviation principle. We show that the good rate function which governs this principle achieves its minimum value at Wigner’s semicircular law, which entails the convergence of the spectral measure to the semicircular law. As a conclusion, we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about Voiculescu’s non commutative entropy.
On a class of II1 factors with at most one Cartan subalgebra
 Ann. of Math
"... Abstract. This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II1. We provide more examples of II1 factors having either zero, one, or several Cartan subalgebras. We also prove a rigidity result for some group measure space II1 fac ..."
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Cited by 87 (8 self)
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Abstract. This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II1. We provide more examples of II1 factors having either zero, one, or several Cartan subalgebras. We also prove a rigidity result for some group measure space II1 factors.
Applications of free entropy to finite von Neumann algebras
, 1997
"... We apply Voiculescu’s free entropy to show that free group factors do not possess a simple maximal abelian selfadjoint subalgebra, hence answering a longstanding question of Ambrose and Singer. Introduction. Our goal in this article is to answer a longstanding question concerning the existence of ..."
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Cited by 68 (3 self)
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We apply Voiculescu’s free entropy to show that free group factors do not possess a simple maximal abelian selfadjoint subalgebra, hence answering a longstanding question of Ambrose and Singer. Introduction. Our goal in this article is to answer a longstanding question concerning the existence of “simple” maximal abelian selfadjoint subalgebras (masa) of factors of type II1. We prove that the factors arising from the free groups on two or more generators possess no such subalgebras. Our techniques for proving this involve, to a very large extent, the brilliant, new free probability
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 62 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
Hochschild Cohomology For Von Neumann Algebras With Cartan Subalgebras
 London Math. Soc. Lecture Note Series 203
"... The main result of this paper is that H n (M;M) = 0, n 1, for von Neumann algebras M with Cartan subalgebras and separable preduals. () Partially supported by an NSF research grant. x1. Introduction The systematic study of the continuous Hochschild cohomology groups H n (M;M), n 1, of a von ..."
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Cited by 51 (16 self)
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The main result of this paper is that H n (M;M) = 0, n 1, for von Neumann algebras M with Cartan subalgebras and separable preduals. () Partially supported by an NSF research grant. x1. Introduction The systematic study of the continuous Hochschild cohomology groups H n (M;M), n 1, of a von Neumann algebra M with coefficients in itself was begun in a series of papers [10, 12, 15, 16] by Johnson, Kadison and Ringrose. Their work was an outgrowth of the KadisonSakai Theorem on derivations, [14, 25], which proved, in an equivalent formulation, that H 1 (M;M) = 0 for all von Neumann algebras. It was natural to conjecture that H n (M;M) = 0 for n 2, and this was settled affirmatively for hyperfinite von Neumann algebras (a class which includes the type I von Neumann algebras) in [16]. The authors established many general results on cohomology in [12, 15, 16], and one important consequence of their work is that it suffices to investigate H n (M;M) when M is type I; II 1 ; I...
About the QWEP conjecture
, 2003
"... This is a detailed survey on the QWEP conjecture and Connes’ embedding problem. Most of contents are taken from Kirchberg’s paper [Invent. Math. 112 (1993)]. ..."
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Cited by 42 (1 self)
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This is a detailed survey on the QWEP conjecture and Connes’ embedding problem. Most of contents are taken from Kirchberg’s paper [Invent. Math. 112 (1993)].
The free entropy dimension of hyperfinite von Neumann algebras
"... ABSTRACT. Suppose M is a hyperfinite von Neumann algebra with a tracial state ϕ and {a1,..., an} is a set of selfadjoint generators for M. We calculate δ0(a1,..., an), the modified free entropy dimension of {a1,..., an}. Moreover we show that δ0(a1,..., an) depends only on M and ϕ. Consequently δ0( ..."
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Cited by 40 (14 self)
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ABSTRACT. Suppose M is a hyperfinite von Neumann algebra with a tracial state ϕ and {a1,..., an} is a set of selfadjoint generators for M. We calculate δ0(a1,..., an), the modified free entropy dimension of {a1,..., an}. Moreover we show that δ0(a1,..., an) depends only on M and ϕ. Consequently δ0(a1,..., an) is independent of the choice of generators for M. In the course of the argument we show that if {b1,..., bn} is a set of selfadjoint generators for a von Neumann algebra R with a tracial state and {b1,..., bn} has finite dimensional approximants, then for any b ∈ R δ0(b1,..., bn) ≥ δ0(b). Combined with a result by Voiculescu this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then δ0(b1,..., bn) = 1. 1.
Connes’ embedding conjecture and sums of Hermitian squares
 Adv. Math
"... Abstract. We show that Connes ’ embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever selfadjoint contraction matric ..."
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Cited by 36 (13 self)
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Abstract. We show that Connes ’ embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever selfadjoint contraction matrices of the same size are substituted for the variables. These algebraic certificates involve sums of hermitian squares and commutators. We prove that they always exist for a similar nonnegativity condition where elements of separable II1factors are considered instead of matrices. Under the presence of Connes’ conjecture, we derive degree bounds for the certificates. 1.