General de Finetti type theorems in noncommutative probability

We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.Comment: This is the final version. Title is changed. to appear in Comm. Math. Phy

Operator valued random matrices and asymptotic freeness

We show that the limit laws of random matrices, whose entries are conditionally independent operator valued random variables having equal second moments proportional to the size of the matrices, are operator valued semicircular laws. Furthermore, we prove an operator valued analogue of Voiculescu's asymptotic freeness theorem. By replacing conditional independence with Boolean independence, we show that the limit laws of the random matrices are Operator-valued Bernoulli laws.Comment: Missing references are adde

Extended de Finetti theorems for boolean independence and monotone independence

We construct several new spaces of quantum sequences and their quantum families of maps in sense of So{\l}tan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by integers, rather than unilateral sequences. In the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: 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. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability.Comment: Comments are welcome! 44pages+reference

Free-Boolean independence for pairs of algebras

We construct pairs of algebras with mixed independence relations by using truncations of reduced free products of algebras. For example, we construct free-Boolean pairs of algebras and free-monotone pairs of algebras. We also introduce free-Boolean cumulants and show that free-Boolean independence is equivalent to the vanishing of mixed cumulants.Comment: Moments-condition for free-Boolean independence is added to Section 4. Title is changed. All comments are welcom

A noncommutative De Finetti theorem for boolean independence

We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. This is a boolean analogue of de Finetti theorem on exchangeable sequences. In the end of the paper, we will discuss the other two invariance conditions which lead to some trivial results.Comment: 30 pages+references, a small result related to faithful states was added to section 7. Many typos are corrected. Any comments are welcome

Free-free-Boolean independence for triples of algebras

In this paper, we introduce the notion of free-free-Boolean independence relation for triples of algebras. We define free-free-Boolean cumulants ans show that the vanishing of mixed cumulants is equivalent to free-free-Boolean independence. A free-free -Boolean central limit law is studied.Comment: 21 pages+ reference. Comments are welcome. arXiv admin note: text overlap with arXiv:1710.0137

First exit and Dirichlet problem for the nonisotropic tempered α\alpha-stable processes

This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $\alpha$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{\tau_D}\right|$ and the first exit time $\tau_D$ are firstly obtained. It is found that the probability density function of $\left|X_{\tau_D}\right|$ or $\tau_D$ exponentially decays with the increase of $\left|X_{\tau_D}\right|$ or $\tau_D$, and $\mathrm{E}\left[\tau_D\right]\sim \left|\mathrm{E}\left[X_{\tau_D}\right]\right|$,\ $\mathrm{E}\left[\tau_D\right]\sim\mathrm{E}\left[\left|X_{\tau_D}-\mathrm{E}\left[X_{\tau_D}\right]\right|^2\right]$. Since $\mathrm{\Delta}^{\alpha/2,\lambda}_m$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator $\mathrm{\Delta}^{\alpha/2,\lambda}_m$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.Comment: 23 pages, 5 figure

Free-Boolean independence with amalgamation

In this paper, we develop the notion of free-Boolean independence in an amalgamation setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence with amalgamation. We also provide a characterization of free-Boolean independence by conditions in terms of mixed moments. In addition, we study free-Boolean independence over a $C^*$-algebra and prove a positivity property.Comment: 24pages+ reference. Comments are welcom

Quantum Observable Generalized Orthoalgebras

Let ${\cal S}(\mathcal{H})$ denote the set of all self-adjoint operators (not necessarily bounded) on a Hilbert space $\mathcal{H}$, which is the set of all physical quantities on a quantum system $\mathcal{H}$. We introduce a binary relation $\bot$ on ${\cal S}(\mathcal{H})$. We show that if $A\bot B$, then $A$ and $B$ are affiliated with some abelian von Neumann algebra. The relation $\bot$ induces a partial algebraic operation $\oplus$ on ${\cal S}(\mathcal{H})$. We prove that $({\cal S}({\mathcal{H}}), \bot, \oplus, 0)$ is a generalized orthoalgebra. This algebra is a generalization of the famous Birkhoff\,--\,von Neumann quantum logic model. It establishes a mathematical structure on all physical quantities on $\mathcal{H}$. In particular, we note that $({\cal S}({\mathcal{H}}), \bot, \oplus, 0)$ has a partial order $\preceq$, and prove that $A\preceq B$ if and only if $A$ has a value in $\Delta$ implies that $B$ has a value in $\Delta$ for every Borel set $\Delta$ not containing $0$. Moreover, the existence of the infimum $A\wedge B$ and supremum $A\vee B$ for $A,B\in \mathcal{S}(\mathcal{H})$ (with respect to $\preceq$) is studied, and it is shown at the end that the position operator $Q$ and momentum operator $P$ in the Heisenberg commutation relation satisfy $Q\wedge P=0$

Remarks on the Sequential Products

In this paper, we show that those sequential products which were proposed by Liu and Shen and Wu in [J. Phys. A: Math. Theor. {\bf 42}, 185206 (2009), J. Phys. A: Math. Theor. {\bf 42}, 345203 (2009)] are just unitary equivalent to the sequential product $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$