A Unified approach to Infinitesimal Freeness with Amalgamation

Operator-Valued Infinitesimal Multiplicative Convolutions

“…Consider an operator-valued infinitesimal probability space pA, B, E, E 1 q and its corresponding upper triangular probability space, the following properties establish the connection between these two spaces (see [15] and [22]).…”

“…(see [15] and [22]). We will omit the subindex in the norm, as it is clear that we use } ¨} r A on upper triangular matrices and } ¨}A on A.…”

“…Then g x is Fréchlet analytic on H `pBq; furthermore, the matrix amplifications pg pmq x q 8 m"1 encodes all possible infinitesimal moments of x (see [22]).…”

“…Curran and Speicher introduced the notion of infinitesimal freeness in the OV framework and they provided an random matrix model which are asymptotically OVI freeness [6]. The detail discussion of OVI freeness was addressed in [22]. In particular, the formula of OVI free additive convolution was also provided.…”

“…A natural question is whether we have OVI analogues for multiplicative convolutions for free, Boolean, and monotone independent random variables. The main purpose of this paper is to answer this question in the affirmative and we apply the method of [22] and [15] to reduce the OVI famework to an 2 ˆ2 upper triangular probability space, for which we have rich tools in the OV realm to deal with them. In this paper, we review some basic knowledge of OV and OVI probability theory in Section 2.…”