Narrow Results

The review of the representation theory of deformations of the CCR is presented. The faithfulness of the Fock representation of q-CCR, twisted CCR and quon CCR is discussed. The more general deformation of CCR is presented. The K₀ and K₁ groups of the twisted CCR algebra are calculated.

In this paper we study the Fock representation of a certain *-algebra which appears naturally in the framework of quantum group theory. It is also a generalization of the twisted CCR-algebra introduced by Pusz and Woronowicz. We prove that the Fock representation is a faithful irreducible representation of the algebra by bounded operators in a Hilbert space, and, moreover, it is the only (up to unitary equivalence) representation possessing these properties.

We consider the $C^{\ast }$-algebra ${{ℰ}}_{n,m}^q$, which is a $q$-twist of two Cuntz–Toeplitz algebras. For the case $\left|q\right|<1$, we give an explicit formula which untwists the $q$-deformation showing that the isomorphism class of ${{ℰ}}_{n,m}^q$ does not depend on $q$. For the case $\left|q\right|=1$, we give an explicit description of all ideals in ${{ℰ}}_{n,m}^q$. In particular, we show that ${{ℰ}}_{n,m}^q$ contains a unique largest ideal ${{ℳ}}_q$. We identify ${{ℰ}}_{n,m}^q/{{ℳ}}_q$ with the Rieffel deformation of ${{𝒪}}_n\otimes {{𝒪}}_m$ and use a K-theoretical argument to show that the isomorphism class does not depend on $q$. The latter result holds true in a more general setting of multiparameter deformations.

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by , where , b_s are the Hida white noise densities. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing . This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise.

We study the holomorphic unitary representations of the Jacobi group based on Siegel–Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel–Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel–Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.

In this paper we study representations associated with the positive solution of a certain difference equation, with initial condition 0. These are representations that arise as Fock representations associated to a quadratic commutation relation. We define a space of parameters for these Fock representations and we determine the regions in these parameter space where the representations are bounded.

The following sections are included:

• Introduction and statement of the problem

• Random variables in CeHeis_ℂ(1)

• Infinite divisibility

• Kernels and matrices

• Positive definite kernels

• Functions of positive definite matrices and kernels

• Conditionally positive definite kernels

• Infinitely divisible kernels

• The Kolmogorov decomposition theorem for ℂ–valued PD kernels

• Boson Fock spaces

• Infinitely divisible kernels and Boson Fock spaces

• References