Infinite Dimensional Analysis, Quantum Probability and Related Topics

We consider the energy representation for the gauge group. The gauge group is the set of C^∞-mappings from a compact Riemannian manifold to a semi-simple compact Lie group. In this paper, we obtain irreducibility of the energy representation of the gauge group for any dimension of M. To prove irreducibility for the energy representation, we use the fact that each operator from a space of test functionals to a space of generalized functionals is realized as a series of integral kernel operators, called the Fock expansion.

For a von Neumann algebra ࡕ acting on a Hilbert space ℋ with a cyclic and separating vector ξ₀, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (ࡕ, ξ₀). For a general bounded Lindblad type generator L of a conservative quantum dynamical semigroup on ࡕ, we give sufficient conditions so that the bounded operator H induced by L via the symmetric embedding of ࡕ into ℋ to be self-adjoint. It turns out that the self-adjoint operator H can be written in the form of a Dirichlet operator associated to a Dirichlet form given in Ref. 23. In order to make the connection possible, we also extend the range of applications of the formula in Ref. 23.

We give an asymptotic result for the occupation of Borel sets of functions by the segments of recurrent Brownian motion on consecutive time intervals [n, n +1], n =0, 1, 2, …. This result provides new information on the behavior of Brownian motion, which is illustrated by examples. A formulation in terms of weak convergence of random measures on Polish space is also given. The proof is based on (a strengthened form of) the Darling–Kac occupation time theorem for Markov chains, and our result can be viewed as a "trajectorial" extension of that theorem. The main role in the occupation limit for Brownian segments is played by the σ-finite Wiener measure, which first appeared in a different context. An extension for segments of symmetric α-stable Lévy processes is also given.

Let M be a von Neumann algebra. We define the noncommutative extension of information geometry by embeddings of M into noncommutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p +1/q =1, we show that we can introduce the α-divergence, for α∈(-1, 1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the α-divergence belongs to the class of quasi-entropies, defined by Petz.

We introduce the forward integral with respect to a pure jump Lévy process and prove an Itô formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Itô formula for the Skorohod integral.

We show how the construction of t-transformation can be applied to the construction of a sequence of monotonically independent noncommutative random variables. We introduce the weakly monotone Fock space, on which these operators act. This space can be derived in a natural way from the papers by Pusz and Woronowicz on twisted second quantization. It was observed by Bożejko that, by taking μ = 0, for the μ-CAR relations one obtains the Muraki's monotone Fock space, while for the μ-CCR relations one obtains the weakly monotone Fock space. We show that the direct proof of the central limit theorem for these operators provides an interesting recurrence for the highest binomial coefficients. Moreover, we show the Poisson type theorem for these noncommutative random variables.

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q≥2 spin values) model on the Cayley tree of order k≥1. The inhomogeneity means that the interaction J_xy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p-adic) Gibbs measures of the model. We show that (i) if q∉pℕ then there is unique Gibbs measure for any k≥1 and ∀ J_xy with | J_xy |< p^{-1/(p -1)}. (ii) For q∈p ℕ, p≥3 one can choose J_xy and k≥1 such that there exist at least two Gibbs measures which are translation-invariant.

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which other dualities, known and new, also find their natural common place.

The standard renormalization procedure consists of introducing a cutoff and then trying to remove it by some limiting procedure. In Ref. 2 a new renormalization technique was introduced based on the idea of renormalizing a closed set of commutation relations and then finding a nontrivial representation for them. In Ref. 5 it was proved that, in the case of quadratic fields the new renormalization procedure leads to quadratic field operator which is gamma distributed in the quadratic vacuum (as one would intuitively expect from the "square" of a white noise) and to Meixner or Pascal distributed Poisson fields. It is natural to ask if the same result can be obtained with the usual cutoff and take-limit procedure. In this paper we prove that the answer to this question is negative. More precisely, we show that, independently of the choice of the cutoff (cf. Sec. 7), if a quadratic field admits a limit in the sense of mixed moments, then this limit will be Gaussian distributed in the vacuum and consequently the associated Poisson fields will have a Poisson distribution.