• Journal of the Korean Mathematical Society
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• v.35 no.3
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• pp.675-690
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• 1998

A role of singletons in quantum central limit theorems is studied. A common feature of quantum central limit distributions, the 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 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.

• Bulletin of the Korean Mathematical Society
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• v.23 no.2
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• pp.117-121
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• 1986

Connes [2] showed that if M is an injective .sigma.-finite factor of type II $I_{1}$ and $B_{\phi}$=C1 for some normal faithful state .phi. on M then M is isomorphic to the Araki-wood factor. In [7], Haagerup has succeeded to show that if M is an injective factor of type II $I_{1}$ with separable predual, then $B_{\phi}$=C1 for every normal faithful state on M. Since injective factors of type II $I_{\lambda}$, 0.leq..lambda.<1, were classified [9], this together classifies all injective factors of type III with separable predual. It is not known for non-injective case. In this paper we consider some conditions under which the bicentralizers be trivial.ivial.