Mixtures of classical and free independence

We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered.
Comment: We rewrote and shortened the earlier version. The third version contains mainly the results which are new compared to the paper of Mlotkowski