Infinite Dimensional Analysis, Quantum Probability and Related Topics

Let X be a right Hilbert C^*-module over A. We study the geometry and the topology of the projective space of X, consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of the p-sphere S_p(X) and the natural fibration , where S_p(X) = {x ∈ X: <x, x> = p}, for p ∈ A a projection. The projective space and the p-sphere are shown to be homogeneous differentiable spaces of the unitary group of the algebra ℒ_A(X) of adjointable operators of X. The homotopy theory of these spaces is examined.

By introducing a color filtration to the multiplicity space , we extend the quantum Itô calculus on multiple symmetric Fock space to the framework of filtered adapted biprocesses. In this new notion of adaptedness, "classical" time filtration makes the integrands similar to adapted processes, whereas "quantum" color filtration produces their deviations from adaptedness. An important feature of this calculus, which we call filtered stochastic calculus, is that it provides an explicit interpolation between the main types of calculi, regardless of the type of independence, including freeness, Boolean independence (more generally, m-freeness) as well as tensor independence. Moreover, it shows how boson calculus is "deformed" by other noncommutative notions of independence. The corresponding filtered Itô formula is derived. Existence and uniqueness of solutions of a class of stochastic differential equations are established and unitarity conditions are derived.

This paper describes a new space, , of complex Wiener distributions for the analysis of multi-parameter generalized stochastic processes . For a certain class of functions and complex Wiener integrals Φ₁, …, Φ_m, F(Φ₁, …, Φ_m) is defined as an element of and its Fock space decomposition determined.

It is shown that the notion of monotone independence introduced by Muraki and Lu leads to an associative product of quantum probability spaces. This product is then used to define monotone Lévy processes on dual semigroups.

In this paper, we shall give the Segal–Bargmann representation space of the one-mode interacting Fock space. For this purpose, we shall define the generalized coherent vector associated with the Szegö–Jacobi parameters and introduce the associated integral transformation.