Construction of the notions of independence in non-commutative probability theory

独立的概念在非对易概率论中很重要。我们构造了一个新的独立性概念(=Q-独立性)，它是一个单参数族，它内插了古典独立性和沃库列斯库的自由独立性。这种构造是基于一族非对易概率空间的Q-积运算。它与Bozejko-Speicher的Q-Fock空间有关。我们构造了中心极限定理、小数定律、卷积和累积量的q-类。我们还发现了一种新的独立概念，我们称之为扭曲的独立。它源于W.Pusz扭曲的正则反对易关系。我们还给出了正自然积(=正泛独立)分类定理的一个简单证明。

Notions of independence are important in non-commutative probability theory (= quantum probability theory). We constructed a new notion of independence (= q-independence) which is a one-parameter family of independence interpolating classical independence and Voiculescu's free independence. The construction is based on the q-product operation for a family of non-commutative probability spaces. It is connected with the q-Fock space of Bozejko-Speicher. We constructed q-analogues of central limit theorem, law of small numbers, convolution and cumulants. We also discovered a new notion of independence which we call twisted independence. It arises from the twisted canonical anti-commutation relations of W. Pusz. We also gave a simple proof for the classification theorem for positive natural products (= positive universal independences).