  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.
