Infinite Dimensional Analysis, Quantum Probability and Related Topics

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples.

We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy⁸). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson.

By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

The paper deals with Lévy processes with values in L₁(H), the Banach space of trace-class operators in a Hilbert space H. Lévy processes with values and parameter in a cone K of L₁(H) are introduced and several properties are established. A family of L₁(H)-valued Lévy processes is obtained via the subordination of K-parameter, L₁(H)-valued Lévy processes, identifying explicitly their generating triplets.

A stochastic integral representation in terms of generalized integral kernel operator is proved for a wide class of quantum martingales which includes regular martingales and the martingales determined by the second quantization of integral operators containing singular kernels.

The main object of this paper is to establish the Hamiltonian of the Hudson–Parthasarathy quantum stochastic differential equation

Muraki has shown that monotonic convolution of probability measures corresponds with composition of the reciprocal Cauchy transforms corresponding with these measures. We provide a new proof of this fact, not involving the combinatorics of moments.

Tracial distributions are defined as the elements of the dual of a countably Hilbertian space and on which a projective limit topology is defined. The connection of these distributions with Hida's generalized functions is discussed.

Coined quantum walks may be interpreted as the motion in position space of a quantum particle with a spin degree of freedom; the dynamics are determined by iterating a unitary transformation which is the product of a spin transformation and a translation conditional on the spin state. Coined quantum walks on ℤ^d can be treated as special cases of coined quantum walks on ℝ^d. We study quantum walks on ℝ^d and prove that the sequence of rescaled probability distributions in position space associated to the unitary evolution of the particle converges to a limit distribution.

We prove the extendability of generalized quantum Markov chains on gauge-invariant parts of UHF algebras to Markov chains on ambient UHF algebras.