Abstract
We present the notions of independence, which appear in non-commutative probability. The basic ones are free, boolean and monotonic independences, formulated for families of algebras indexed by totally ordered set. A generalization of the latter two is the bm-independence, defined for partially ordered index sets. For each independence there is an analogue of the classical central limit theorem. In the case of bm-independence this depends also on the index set. Examples of such partially ordered index sets are discrete lattices in symmetric positive cones. Mathematics subject classification (2010): 46L54, 81R50.
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