## Non-commutative notions of stochastic independence.(English)Zbl 1028.46094

A complete classification of the notions of non-commutative independence is given. To discuss independence in the non-commutative case we have to pass from $$\sigma$$-algebras of events to algebras of functions generated by the indicator functions of events. Probabilities are replaced by expectations which are given by a state $$\Phi$$ (a positive normalized linear functional) on $$A$$. Let $$A$$ be the algebra generated by algebras $$A_1$$ and $$A_2$$. We say that $$A_1$$ and $$A_2$$ are independent if $$\Phi$$ can be computed for its restrictions $$\varphi_1= \Phi|_{A_1}$$ and $$\varphi_2= \Phi|_{A_2}$$ by applying a universal prescription. Universal prescriptions are given by universal products. There are four conditions (P1)–(P4) on $$\varphi_1$$, $$\varphi_2$$, implying independence in the commutative case. Definitions of three different universal products in the case of algebras are given: the tensor products (H), free products (V) and Boolean products (W). These products correspond to tensor, free and Boolean kinds of independence, respectively. The main results of the paper:
Theorem 3 (Unital case). There are exactly two universal products satisfying (P1)–(P4): one given by the tensor case (H) and one given by the free case (V).
Theorem 4 (General case). There are three families of non-degenerate universal products satisfying (P1)–(P4), each family being parametrized by $$q\in\mathbb{C}^* =\mathbb{C}\setminus \{0\}$$. The three families are called the tensor family, the free family and the Boolean family. For $$q\in \mathbb{C}^*$$ the corresponding member of the tensor, free and Boolean family is given by $$\varphi_1\varphi_2= q((q^{-1}\varphi_1) \bullet(q^{-1} \varphi_2))$$, where “$$\bullet$$” denotes the tensor, free and Boolean product, respectively.

### MSC:

 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras
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