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Operator-valued free Fisher information and modular frames


Authors: Bin Meng, Maozheng Guo and Xiaohong Cao
Journal: Proc. Amer. Math. Soc. 133 (2005), 3087-3096
MSC (2000): Primary 46L54, 42C15
DOI: https://doi.org/10.1090/S0002-9939-05-08111-6
Published electronically: April 25, 2005
MathSciNet review: 2159789
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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the operator-valued free Fisher information for a random variable in an operator-valued noncommutative probability space and point out its relations to the amalgamated freeness. Using M. Frank and D. Larson's modular frame notion we can construct the conjugate variable for an operator-valued semicircle variable with conditional expectation covariance. Then we obtain its free Fisher information and show it is equal to the index of the conditional expectation. At last the conjugate variable with respect to a modular frame operator for a semicircle variable is also constructed.


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Additional Information

Bin Meng
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Address at time of publication: College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China
Email: b.meng@nuaa.edu.cn

Maozheng Guo
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Xiaohong Cao
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-05-08111-6
Keywords: Conjugate variable, free Fisher information, Hilbert $C^\ast$-module, modular frame
Received by editor(s): February 12, 2004
Received by editor(s) in revised form: June 7, 2004, and June 8, 2004
Published electronically: April 25, 2005
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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