Operator-Valued Moment Series of the Generating Operator of L(F₂) 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.