On Certain Free Product Factors via an Extended Matrix Model

Dykema, Ken

doi:10.1006/jfan.1993.1025

Cited by 93 publications

(99 citation statements)

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“…An elegant proof of Theorem 5.4.2 for Gaussian matrices which avoid combinatorial arguments appears in [CaC04]. Theorem 5.4.2 was extended to non Gaussian entries in [Dyk93b]. The proof of Theorem 5.4.10 we presented follows the characterization of the law of free unitary variables by a Schwinger-Dyson equation given in [Voi99, Proposition 5.17] and the ideas of [CoMG06].…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…Let U nc n be the universal C * -algebra generated by the entries of a n×n unitary matrix. By combining the formula U nc n,red ⊗ M n = M n * red C(T) of McClanahan [12] with the formula M n * W * (F s ) = W * (F n 2 s ) ⊗ M n of Dykema [7] we obtain U nc,"…”

Section: Proof (1) Let U V Be the Fundamental Representations Of A mentioning

confidence: 99%

Le Groupe Quantique Compact Libre U(n)

Banica

1997

Communications in Mathematical Physics

145

334

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Abstract. The free analogues of U (n) in Woronowicz's compact quantum group theory are the quantum groups {A u (F )|F ∈ GL(n, C)} introduced by Van Daele and Wang. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If FF ∈ RI n we find an embeddingis the deformation of SU (2) that we previously studied. We use the representation theory and Powers' method for showing that the reduced algebras A u (F ) red are simple, with at most one trace.

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“…Using the matrix model, Voiculescu showed that if (y ij ) 1≤i,j≤N is a * -free family of circular elements in a von Neumann algebra M with respect to a normal faithful state φ, then the matrix y = t k √ 4 − t 2 ) and where u and b are * -free. These and results of a similar nature have been instrumental in applications of free probability to the study of the free group factors L(F n ) and related factors; some of the first of these were [14], [11], [4], [12], [5].…”

Section: Introductionmentioning

confidence: 99%