A three-state independence in non-commutative probability

Understand

We define a new independence in non-commutative probability, called $\alpha$-freeness, with respect to a triplet of states.

• This concept unifies several independences in non-commutative probability, in particular, free, monotone, antimonotone and Boolean ones as well as conditionally free, conditionally monotone and conditionally antimonotone independences.
• Moreover, the associative law of $\alpha$-freeness is transferred to the other independences.
• As a consequence, $\alpha$-free cumulants unify the cumulants for free, monotone, antimonotone and Boolean independences.