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**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.

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.

Reviewer: Victor Sharapov (Volgograd)