R-transform Theory On a Tower of Amalgamated Noncommutative Probability Spaces

In [9], we observed Amalgamated R-transform Theory. Different from the original definition of Voiculescu and Speicher, we define R-transforms of operator-valued random variable(s) by operator-valued formal series. By doing that we can establish Amalgamated R-transform calculus. In this paper, we will concentrate on observing R-transforms on a tower of amalgamated noncommutative probability spaces. We will consider the R-transform calculus and relation between R-transforms on the tower and distributions with respect to the tower. As application, we will observe the compatibility of the given tower and commuting ladders of amalgamated noncommutative probability spaces