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

In this paper, we will consider R-transform theory and R-transform calculus for compatible noncommutative probability space and amagamated noncommutative probability space. By doing this, we can realize the relation between scalar-valued R-transforms and operator-valued moment series, under the compatibility. Also, we can see that there is a big gap between freeness and amalgamated freeness.Comment: 37 page

Moment Series and R-transform of the Generating Operator of L(FN)L(F_{N})

In this paper, we will consider the free probability theory of free group factor $L(F_{N}),$ where $F_{k}$ is the free group with $k$-generators. We compute the moment series and the R-transform of the generating operator, $T=g_{1}+...+g_{N}+g_{1}^{-1}+..+g_{N}^{-1}.$Comment: 12 page

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=\QTR{cal}{C}^{N}, for N\in \QTR{Bbb}{N}, induced by a (scalar-valued) noncommutative probability space, (A,\QTR{cal}{\phi}), with a suitable $B$-functional, $E:T\to B,$ defined by $\phi .$ On this framework, we will observe amalgamated R-transform calculus with respect to a $B$-functional, $E.$ The technique and idea are came from those in [12] and Free Probability of type $B$ studied in [31].Comment: 46 page

Graph Free Product of Noncommutative Probability spaces

In this paper, we will construct the graph free product of noncommutative probability space. This is the attempt to explain and observe the combinatorial-object-depending probabilistic structure

Operator-Valued Moment Series of the Generating Operator of L(F_2) Over the Commutator Group von Neumann algebra L(K)

In this paper, we will consider the generating operator of the free group factor L(F_2). Then we can construct the group von Neumann algebra L(K), where K is the commutator group of F_2 and the conditional expectation E. Then (L(F_2), E) is the W*-probability space with amalgamation over L(K). In this paper, we will compute the trivial operator-valued moment series of the generating operator of L(F_2) over L(K). This computation is the good example for studying the operator-valued distribution, since the operator-valued moment series of operator-valued random variables contain algebraic and combinatorial free probability information about the opeartor-valued distributions.Comment: 17 page

Moments of the Generating Operator of Two Copies of L(F_2)'s over L(F_1)

064In this paper, we will consider an example of a (scalar-valued) moment series, under the compatibility. Suppose that we have an amalgamated free product of free group algebras, $L(F_{2})*_{L(F_{1})}L(F_{2})=L()*_{L()}L()$ 064We will provide the method how to find the moment series of $a+b+a^{-1}+b^{-1}+c+d+c^{-1}+d^{-1}.$ Amalgamated freeness of $a+b+a^{-1}+b^{-1}$ and $c+d+c^{-1}+d^{-1}$ over $L(F_{1})$ is used and some combinatorial functions (to explain the recurrence relations) are used to figure out the $n$-th moment of this element.Comment: 63 pages, It is presented in COSy 200

Compressed Random Variables in a Graph W*-Probability Space

In this paper, we will consider the free probabilistic information about compressed random variables in a graph W*-Probability space. Recall the diagonal compressed random variables in a graph W*-probability space. In particular, we can see that the free moments and cumulants of the fixed compressed random variable of a random variable x are exactly same as the diagonal compressed part of x

Amalgamated R-diagonal Pairs

In this paper, we will consider the properties of amalgamated R-diagonal pairs. We characterize the amalgamated R-diagonality of pairs of amalgamated random variables by certain cumulant-relation.Comment: 16 page

Random Variables in Graph W*-Probability Spaces

In [16], we observed the graph W*-probability theory. In this paper, we will review [16] and introduce special amalgamated random variables in this amalgamated W*-probability space. In particular, we will observe the amalgamated semicircularity, amalgamated evenness and amalgamated R-diagonality. As an example, we will compute the trivial moments and trivial cumulants of the generating operator of the graph W*-algebra.Comment: 38 page

Graph W*-probability Theory

In this paper, we will consider the graph w*-probability theory.Comment: 60 page