Infinite Dimensional Analysis, Quantum Probability and Related Topics

Several results giving upper bounds for the speed of convergence in non-commutative central limit theorems are known. A natural question is whether there also exist lower bounds, or whether there exist classes of probability measures where the speed of convergence is faster. In this paper, we answer this for the Boolean central limit theorem in the bounded support case and using the Lévy distance. We show that for a probability measure $\mu$ of bounded support and nonzero third moment the rate of convergence is at least $2^{-1}|m_3(\mu )|n^{-1/2}+O(n^{-1})$, and when we have zero third moment, then the rate of convergence is $2^{-1}(m_4(\mu )-1)n^{-1}+O(n^{-3/2})$. To achieve this, we improve a previous result concerning the position and weights of two atoms that eventually appear in the converging measure. The proofs only require elementary techniques of complex analysis.

In this paper, we introduce a quantum decomposition of a two-dimensional Bernoulli random variable $({\xi }_1,{\xi }_2)$, where $E({\xi }_1)=E({\xi }_2)=0$, $E({\xi }_1^2)=E({\xi }_2^2)=1$ and $E({\xi }_1{\xi }_2)=c\in (-1,1)$. Based on this quantum decomposition, we defined the two-dimensional quantum Bernoulli process and set a corresponding central limit theorem, both in the sense of quantum moments and in the sense of characteristic function.

Local actions of ${\mathbb{P}}_{\mathbb{N}}$, the group of finite permutations on $\mathbb{N}$, on quasi-local algebras are defined and proved to be ${\mathbb{P}}_{\mathbb{N}}$-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of $C^{*}$-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon ${\mathbb{P}}_{\mathbb{N}}$ in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.

In this paper, we study stochastic currents of Brownian motion $\xi (x)$, $x\in {ℝ}^d$, by using white noise analysis. For $x\in {ℝ}^d\setminus \{0\}$ and for $x=0\in ℝ$ we prove that the stochastic current $\xi (x)$ is a Hida distribution. Moreover for $x=0\in {ℝ}^d$ with $d>1$ we show that the stochastic current is not a Hida distribution.

We study criteria which ensure that Gibbs states (often also called generalized vacuum states) on distance-regular graphs are positive. Our main criterion assumes that the graph can be embedded into a growing family of distance-regular graphs. For the proof of the positivity we then use polynomial hypergroup theory and translate this positivity into the problem whether for $x\in [-1,1]$ the function $n\mapsto x^n$ has a positive integral representation w.r.t. the orthogonal polynomials associated with the graph. We apply our criteria to several examples. For Hamming graphs and the infinite distance-transitive graphs we obtain a complete description of the positive Gibbs states.

A commuting tuple of operators $(S_1,\dots ,S_{n-1},P)$, defined on a Hilbert space ${\mathscr{H}}$, for which the closed symmetrized polydisc