Compatibility of a noncommutative probability space and a noncommutative probability space with amalgamation

摘要

In this paper, we will consider R-transform theory and R-transform calculusfor compatible noncommutative probability space and amagamated noncommutativeprobability space. By doing this, we can realize the relation betweenscalar-valued R-transforms and operator-valued moment series, under thecompatibility. Also, we can see that there is a big gap between freeness andamalgamated freeness.