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

独立性的概念在非对易概率论（=量子概率论）中很重要。我们构造了一个新的独立性概念（= q-独立性），它是插值经典独立性和Voiculescu自由独立性的单参数独立性族。该构造基于非交换概率空间族的q-乘积运算。它与Bozejko-Speicher的q-Fock空间相连。我们构造了中心极限定理、小数定律、卷积和累积量的q-模拟。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的扭曲正则反对易关系。娘们我们还给出了正自然积分类定理（=正泛独立性）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).