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September 2004 Operator-valued Free Entropy and Modular Frames
Maozheng Guo, Bin Meng, Xiaohong Cao
Methods Appl. Anal. 11(3): 331-344 (September 2004).


We introduce the operator-valued relative free entropy $\chi_\mb^{\ast}(X_1,X_2,\cdots,X_n:\mb)$ of a family of self-adjoint random variables $X_1,X_2,\cdots,X_n$ in a $\mb$-valued noncommutative probability space $(\ma,\emb,\mb)$. This notion extends D. Voiculescu's relative free entropy $\Phi^{\ast}$ which defined in a tracial W*-noncommutative probability space to a more general context. The free entropy of a semicircular variable with conditional expectation covariance can be computed by using the modular frames and then we point out the relation between the free entropy of a semicircular variable and the index of a conditional expectation. At last, we obtain an estimate of the free entropy dimension $\delta^\ast_{\mb,\tau}$.


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Maozheng Guo. Bin Meng. Xiaohong Cao. "Operator-valued Free Entropy and Modular Frames." Methods Appl. Anal. 11 (3) 331 - 344, September 2004.


Published: September 2004
First available in Project Euclid: 11 May 2006

zbMATH: 1100.46038
MathSciNet: MR2214678

Rights: Copyright © 2004 International Press of Boston

Vol.11 • No. 3 • September 2004
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