Graph W ∗-probability on the Free Group Factor L(fn)

In this paper, we will consider the free probability on the free group factor L(F N), in terms of Graph W *-probability, where F k is the free group with k-generators. The main result of this paper is to reformulate the moment series and the R-transform of the operator T 0 = N j=1 (x j) , where x j 's are free semicircular elements, j = 1, ..., N, generating L(F N), by using the Graph W *-probability technique. To do that, we will use the graph W *-probability technique to find the moments and cumulants of the identically distributed random variable T with T 0. This will be a good example of an application of Graph W *-Probability Theory. We also can see how we can construct the W *-subalgebra which is isomorphic to the free group factor L(F N) in the graph W *-probability space (W * (G), E) .