An analogue of Hunt's representation theorem in quantum probability

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of, instruments on groups and the associated semigroups of probability operators, which now are defined on spaces of functions with values in a von Neumann algebra. We consider a semigroup of probability operators with a continuity property weaker than uniform continuity, and we succeed in characterizing its infinitesimal generator under the additional hypothesis that twice differentiable functions belong to the domain of the generator. Such hypothesis can be proved in some particular cases. In this way a partial quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. Our result provides also a closed characterization of generators of a new class of not norm continuous quantum dynamical semigroups.     

Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

The aim of this note is to characterize certain probability laws on a class of quantum groups or braided groups that will be called nilpotent. First, we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as a standard example. We determine functional on the braided line and on this group satisfying an analogue of the Bernstein property (see [3]). i.e. that the sum and difference of independent Gaussian random variables are also independent. We also study continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend to nilpotent quantum groups and braided groups recent results proving the uniqueness of the embedding of an infinitely divisible probability law in a continuous convolution semigroup for simply connected nilpotent Lie groups.     

Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

Summary. The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg–Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that the sum and difference of independent random variables are also independent. The corresponding functionals on the braided line, braided plane and a braided q-Heisenberg–Weyl group are determined. Section 5 deals with continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend recent results proving the uniqueness of the embedding of an infinitely divisible probability law into a continuous convolution semigroup for simply connected nilpotent Lie groups to nilpotent quantum groups and braided groups. Finally, in Section 6 we give some indications how the semigroup approach of Heyer and Hazod to the Bernstein theorem on groups can be extended to quantum groups and braided groups. Received: 30 October 1996 / In revised form: 1 April 1997     

Probability operators and convolution semigroups of instruments in quantum probability

Summary In quantum measurement theory a central notion is that of instrument, which is a certain kind of operator-valued measure. In this paper instruments on locally compact groups are studied and, as in classical probability theory, probability operators associated with instruments are introduced. Then, the generator of a norm continuous semigroup of probability operators is characterized.     

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Some <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> properties of semigroups of measures on Lie groups

We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L ²-space of Haar measure. Necessary and sufficient conditions are given for the infinitesimal generator to be self-adjoint and the associated symmetric Dirichlet form is constructed. We show that the generated Markov semigroup is trace-class if and only if the measures have a square-integrable density. Two examples are studied in some depth where the spectrum can be explicitly computed, these being the n-torus and Riemannian symmetric pairs of compact type.     

Lie algebras and recurrence relations I

We study representations of the Heisenberg-Weyl algebra and a variety of Lie algebras, e.g., su(2), related through various aspects of the spectral theory of self-adjoint operators, the theory of orthogonal polynomials, and basic quantum theory. The approach taken here enables extensions from the one-variable case to be made in a natural manner. Extensions to certain infinite-dimensional Lie algebras (continuous tensor products, q-analogs) can be found as well. Particularly, we discuss the relationship between generating functions and representations of Lie algebras, spectral theory for operators that lead to systems of orthogonal polynomials and, importantly, the precise connection between the representation theory of Lie algebras and classical probability distributions is presented via the notions of quantum probability theory. Coincidentally, our theory is closed connected to the study of exponential families with quadratic variance in statistical theory.     

紧拓扑半群上概率测度卷积序列的极限性质

本文讨论紧拓扑半群上概率测度卷积序列的若干重要极限性质．在第１节中，我们讨论测度集的代数结构与其支撑集代数结构的关系．第２节的定理１，通过支撑集的代数结构给出组合收敛测度序列的一个极限定理．在定理２中我们讨论独立同分布时的情形，建立了一类紧半群上的Ｋａｗａｄａ－Ｉｔ型结果．这些定理推广了紧群、紧交换半群、紧Ｌ－Ｘ半群上一些相应的结论．     

Continuous state branching semigroups

Summary This paper is concerned with Markov processes with continuous creation where the phase space is a general separable compact metric space. The transition probabilities for such a process determine a semigroup of operators acting on a function space over the collection of bounded Borel measures on the phase space. Such a semigroup is characterized by a particular convolution condition and is called a continuous state branching semigroup. A connection is established between continuous state branching semigroups and certain semigroups of nonlinear operators and then this connection is exploited to establish an existence theorem for the former.Research associated with a project in probability at Princeton University supported by the Office of Army Research.     

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b ₂-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.     

Trasformate di Radon e operatori di convoluzione su gruppi e algebre di Lie

The following is an expository paper concerning the relations between Radon transforms and convolution operators associated to singular measures. A quick review of the classical theorems is presented and a recent result of F. Ricci and the author in the framework of compact Lie groups and Lie algebras is outlined.     

Mehler semigroups,Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.     

Intertwined evolution operators

We construct a convolution algebra of admissible homomorphisms defined on a ‘test space’ to demonstrate the fundamental role of convolution in the study of intertwined evolution operators of linear ordinary differential equations in Banach spaces and probability theory. The choice of test space makes the framework we present quite versatile. The applications include semigroups of linear operators, empathy, integrated semigroups and empathies and the convolution semigroups of probability theory.     

Riesz Potentials and Liouville Operators on Fractals

An analogue to the theory of Riesz potentials and Liouville operators in R ⁿ for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of Euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved.     

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.     

On a Class of Non-Translation Invariant Feller Semigroups on Lie Groups

We consider a class of Feller semigroups on Lie groups which fail to commute with left translation due to the existence of a cocycle h which is identically one for Lévy processes. Under certain conditions, we are able to show that the infinitesimal generator of such a semigroup has the Lévy–Khintchine–Hunt form but with variable characteristics, thus we obtain an extension of classical work in Euclidean space by Courrège.     

Convolution solutions of an abstract stochastic Cauchy problem

We study convolution solutions of an abstract stochastic Cauchy problem with the generator of a convolution operator semigroup. In the case of additive noise, we prove the existence and uniqueness of a weak convolution solution; this solution is described by a formula generalizing the classical Cauchy formula in which the solution operators of the homogeneous problem are replaced by the convolution solution operators of the homogeneous problem. For the problem with multiplicative noise, we find a condition under which the weak convolution solution coincides with the soft solution and indicate a sufficient condition for the existence and uniqueness of a weak convolution solution; the latter can be obtained by the successive approximation method.     

On Racah operators

As a generalization of the well-known Racah coefficients (defined for finite-dimensional representations of semisimple Lie groups), we introduce the notion of Racah operators for locally compact groups with “nice” dual space. In the case of the group PSL(2,?), these operators are explicitly indicated.     

Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group. Research Supported by NSERC of Canada Grant No. A8227.     

Some Results Concerning Convergence of Convolution Products of Probability Measures on Discrete Semigroups

Two types of conditions have been significant when considering the convergence of convolution products of nonidentical probability measures on groups and semigroups. The essential points of a sequence of measures have been useful in characterizing the supports of the limit measures. Also, enough mass eventually on an idempotent has proven sufficient for convergence in a number of structures. In this paper, both of these types of conditions are analyzed in the context of discrete non-abelian semigroups. In addition, an application to the convergence of nonhomogeneous Markov chains is given.