# A Unified approach to Infinitesimal Freeness with Amalgamation

## TL;DR

This paper explores infinitesimal freeness in operator-valued probability, establishing equivalences with matrix-based frameworks, introducing cumulants, and deriving convolution formulas for free additive and multiplicative operations.

## Contribution

It introduces the notion of operator-valued infinitesimal cumulants and shows their role in characterizing infinitesimal freeness, connecting it to matrix-based frameworks and convolution formulas.

## Key findings

- OVI freeness is equivalent to free independence over 2x2 upper triangular matrices.
- Introduces OVI cumulants and characterizes freeness via their vanishing.
- Provides formulas for free additive and multiplicative convolutions in OVI setting.

## Abstract

We consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of $2\times 2$ upper triangular matrices. We introduce the notion of OVI cumulants and investigate its properties, and we then deduce that the OVI freeness is equivalent to the vanishing of our mixed cumulants. Moreover, we derive the formula for obtaining the free additive and multiplicative convolutions within the realm of OVI freeness.