Several Problems In Non-commutative Probability Spaces

The contents of this thesis are divided into three parts.1. Several topics in infinitesimal non-commutative probability spacesFirst, the R-transform, semicircular element and Poisson element in infinitesimal non-commutative probability spaces are defined, their basic propositions are given. Secondly, central limit theorem in multi-dimensions is proved by means of R-transform, Poisson limit theo-rem and limit theorem for the triangular arrays are also proved. Finally, even element and R-diagonal element are defined, their relation and some arithmetic distributions are given.2. Non-crossing linked partitions on finite totally ordered setFirst, we proved that the set of all non-crossing linked partitions on finite totally ordered set (NCL(n)) is a lattice under partial order≤nc. Secondly, an algebraically equivalent description of the connected component of non-crossing linked partition is given by means of permutation groups.3. Non-crossing linked partitions of type BFirst, linked partitions of type B (LP(B)(n)) and non-crossing linked partitions of type B (NCL,(B)(n)) are defined, their basic propositions are given. Secondly, two basic operations on LP(B)(n) are defined, and using them we study the relation between NCL(B)(n) and non-crossing partitions of type B (NC(B)(n)). Then we generalize partial order≤nc to NCL(B)(n) and prove NCL(B)(n) is not a lattice under≤nc.Finally we obtain some enumerative results of NCL(B)(n) and give an algebraically equivalent description of the connected component of non-crossing linked partitions of type B by means of permutation groups.