Open AccessOperator-valued free Fisher information and modular frames
Author(s) -
Bo Meng,
Maozheng Guo,
X. Cao
Publication year - 2005
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/s0002-9939-05-08111-6
Subject(s) - mathematics , operator (biology) , noncommutative geometry , variable (mathematics) , random variable , discrete mathematics , pure mathematics , statistics , mathematical analysis , biochemistry , chemistry , repressor , transcription factor , gene
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.