We will discuss general Central Limit Theorems in non-commutative probability.
Broadly, a non-commutative probability space is a pair (A, E), where A is a unital algebra whose elements are interpreted as "non-commutative random variables" and E a unital linear functional on A playing the role of "expectation". (For example, take a matrix algebra and the normalized trace.) We will review the Central Limit Theorem of Speicher for non-commutative random variables that satisfy a certain natural class of commutation relations, present a recent generalization, and discuss some interesting implications for non-commutative Gaussians.