Journal of the
Korean Mathematical Society JKMS

A role of singletons in quantum central limit theorems is studied. A common feature of quantum central limit distributions, the {\it singleton condition} which guarantees the symmetry of the limit distributions, is revisited in the category of discrete groups and monoids. Introducing a general notion of quantum independence, the {\it singleton independence} which include the singleton condition as an extremal case, we clarify the role of singletons and investigate the mechanism of arising non-symmetric limit distributions.

Keywords: quantum central limit theorem, singleton, discrete group, monoid, minimal generators, Haagerup state, free group, independence

MSC numbers: Primary 60A05; Secondary 60F17, 81Q99