A three-state independence in non-commutative probability

Abstract: 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. The central limit theorem for $\alpha$-freeness is computed. The limit distribution turns out to be a triplet of the Kesten distributions.