UC Berkeley

The main theme of this thesis is to develop de Finetti type theorems in noncommutative probability. In noncommutative area, there are independence relations other than classi- cal independence, e.g. Voiculescu’s free independence, Boolean independence and Muraki’s monotone independence. Free analogues of de Finetti type theorems were discovered by Ko ̈stler and Speicher and were developed by Banica, Curran and Speicher. Here, we will define noncommutative distributional symmetries for Boolean and monotone independence and we will prove de Finetti type theorems for them. These distributional symmetries are defined via coactions of quantum structures including Woronowicz C∗-algebra and So ltan’s quantum families of maps. We show that the joint distribution of an infinite sequence of noncommutative random variables satisfies boolean exchangeability is equivalent to the fact that the sequence of the random variables is identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. Then, we define noncom- mutative versions of spreadability and show Ryll-Nardzewski type theorems for monotone independence and boolean independence. We will show that, roughly speaking, an infinite bilateral sequence of random variables is monotonically(boolean) spreadable if and only if the variables are identically distributed and monotone(boolean) with respect to the conditional expectation onto its tail algebra. In the end of this thesis, we will prove general de Finetti theorems for classical, free and boolean independence. Our general de Finetti theorems work for non-easy quantum groups, which generalizes a recent work of Banica, Curran and Spe- icher. For infinite sequences, we determine maximal distributional symmetries which means the corresponding de Finetti theorem fails if the sequence satisfies more symmetries other than the maximal one.