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R-Diagonal Elements and Freeness With Amalgamation

Published online by Cambridge University Press:  20 November 2018

Alexandru Nica ,
Dimitri Shlyakhtenko  and
Roland Speicher
Alexandru Nica
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1, e-mail: anica@math.uwaterloo.ca
Dimitri Shlyakhtenko
Affiliation:
Department of Mathematics University of California at Los Angeles Los Angeles, California 90095-1555 U.S.A., e-mail: shlyakht@math.ucla.edu
Roland Speicher
Affiliation:
Institut für Angewandte Mathematik Universität Heidelberg, Germany
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Abstract

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The concept of $R$-diagonal element was introduced in [5], and was subsequently found to have applications to several problems in free probability. In this paper we describe a new approach to $R$-diagonality, which relies on freeness with amalgamation. The class of $R$-diagonal elements is enlarged to contain examples living in non-tracial $*$-probability spaces, such as the generalized circular elements of [7].

Keywords

46L54
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 2 , 01 April 2001 , pp. 355 - 381
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Nica, A., R-transforms of free joint distributions and non-crossing partitions. J. Funct. Anal. 135 (1996), 271–296.Google Scholar
[2] Nica, A., Shlyakhtenko, D. and Speicher, R., Some minimization problems for the free analogue of the Fisher information. Adv. in Math. 141 (1999), 282–321.Google Scholar
[3] Nica, A., Shlyakhtenko, D. and Speicher, R., Maximality of the microstates free entropy for R-diagonal elements. Pacific J. Math. 187 (1999), 333–347.Google Scholar
[4] Nica, A. and Speicher, R., A “Fourier transform” for multiplicative functions on non-crossing partitions. J. Algebraic Combin. 6 (1997), 141–160.Google Scholar
[5] Nica, A. and Speicher, R., R-diagonal pairs—a common approach to Haar unitaries and circular elements. In: Free Probability Theory (ed. D. V. Voiculescu), Fields Inst. Comm. 12 (1997), 149–188.Google Scholar
[6] Nica, A. and Speicher, R., Commutators of free random variables. Duke Math. J. 92 (1998), 553–592.Google Scholar
[7] Shlyakhtenko, D., Free quasi-free states. Pacific J. Math. 177 (1997), 329–368.Google Scholar
[8] Shlyakhtenko, D., Some applications of freeness with amalgamation. J. Reine Angew.Math. 500 (1998), 191–212.Google Scholar
[9] Speicher, R., Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 (1994), 193–206.Google Scholar
[10] Speicher, R., Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Amer.Math. Soc. 132(1998), x+88.Google Scholar
[11] Voiculescu, D. V., Circular and semicircular systems and free product factors. In: Operator algebras, unitary representations, enveloping algebras, and invariant theory, Progress in Math. 92, (eds., A. Connes et al.), Birkhäuser, Boston, 1990, 45–60.Google Scholar
[12] Voiculescu, D. V., The analogues of entropy and of Fisher's information measure in free probability theory, VI: Liberation and mutual free information. Adv. in Math. 146 (1999), 101–166.Google Scholar
[13] Voiculescu, D. V., Dykema, K. J. and Nica, A., Free random variables. CRM Monograph Series 1, Amer. Math. Soc., Providence, 1992.Google Scholar