Amalgamated R-transform Theory on Toeplitz-Probability Spaces over Toeplitz Matricial Algebras

In this paper, we will consider a noncommutative probability space, $(T,E),$ over a Toeplitz matricial algebra $B=\text{QTR}cal{C}^N,$ for $N\in \QTR{Bbb}{N}$,inducedbya(scalar-valued)noncommutativeprobabilityspace,$(A,\QTR{cal}{\phi}),$withasuitable$B$-functional,$E:T\to B,$definedby$\phi .$ On this framework, we will observe amalgamated R-transform calculus with respect to a $B$-functional, $E\mathrm{.}$ The technique and idea are came from those in [12] and Free Probability of type $B$ studied in [31].