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Classical Bargmann’s representation is given by operators acting on the space of holomorphic functions with scalar product $〈z^n,z^k{〉}_q={\delta }_{n,k}{\left[n\right]}_q!=F\left(z^n{\overline{z}}^k\right)$. We consider the problem of representing the functional F as a measure. We prove the existence of such a measure for q > 1 and investigate some of its properties like uniqueness and radiality.

We study a certain class of von Neumann algebras generated by selfadjoint elements ${\omega }_i=a_i+a{⁺}_i$, where $a_i,a{⁺}_i$ satisfy the general commutation relations: $a_{i}a{⁺}_j={\sum }_{r,s}{t}_j^i{r}_{s}a{⁺}_r{a}_s+{\delta }_{ij}Id$. We assume that the operator T for which the constants $t_j^i{r}_s$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the number of generators...

The classical Bargmann representation is given by operators acting on the space of holomorphic functions with the scalar product $⟨zⁿ|z^k{⟩}_q={\delta }_{n,k}{\left[n\right]}_q!=F\left(zⁿz{̅}^k\right)$. We consider the problem of representing the functional F as a measure for q > 1. We prove the existence of such a measure and investigate some of its properties like uniqueness and radiality. The above problem is closely related to the indeterminate Stieltjes moment problem.

We present a new proof of Janson’s strong hypercontractivity inequality for the Ornstein-Uhlenbeck semigroup in holomorphic algebras associated with CAR (canonical anticommutation relations) algebras. In the one generator case we calculate optimal bounds for t such that $U_t$ is a contraction as a map $L₂\left(\right)\to L_p\left(\right)$ for arbitrary p ≥ 2. We also prove a logarithmic Sobolev inequality.