Japanese Journal of Mathematics

, 3:163

Aspects of free analysis


DOI: 10.1007/s11537-008-0753-4

Cite this article as:
Voiculescu, DV. Jpn. J. Math. (2008) 3: 163. doi:10.1007/s11537-008-0753-4


We survey the analysis around the free difference quotient derivation, which is the natural derivation for variables with the highest degree of noncommutativity. The analogue of the Fourier transform is then bialgebra duality for the bialgebra with derivation-comultiplication to which the free difference quotient gives rise and which involves fully matricial analytic functions. Some of the motivation from free probability, especially free entropy and random matrices are also discussed.

Keywords and phrases:

free difference quotientgeneralized resolventsinfinitesimal bialgebrasfree entropy

Mathematics Subject Classification(2000):

16W3046L54 (primary)47A5615A52 (secondary)

Copyright information

© The Mathematical Society of Japan and Springer Japan 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA