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Bargmann representation of q-commutation relations for q > 1 and associated measures

100%
Królak I.
Banach Center Publications
|
2007
|
tom 78
|
nr 1
171-183
EN
The classical Bargmann representation is given by operators acting on the space of holomorphic functions with the scalar product $⟨zⁿ|z^k⟩_q = δ_{n,k}[n]_q! = F(zⁿz̅^k)$. 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.
2
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Optimal Holomorphic Hypercontractivity for CAR Algebras

100%
Królak I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2010
|
tom 58
|
nr 1
79-90
EN
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 $L₂(𝓗) → L_p(𝓗)$ for arbitrary p ≥ 2. We also prove a logarithmic Sobolev inequality.
3
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Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case

100%
Królak I.
Banach Center Publications
|
2006
|
tom 73
|
nr 1
277-284
EN
We study a certain class of von Neumann algebras generated by selfadjoint elements $ω_i = a_i + a⁺_i$, where $a_i, a⁺_i$ satisfy the general commutation relations: $a_ia⁺_j = ∑_{r,s} t^{i}_{j}^{r}_{s} a⁺_r a_s + δ_{ij}Id$. We assume that the operator T for which the constants $t^{i}_{j}^{r}_{s}$ 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 our algebra is a factor. The result was obtained for q-commutation relations by P. Śniady [Snia] and recently by E. Ricard [R]. The latter proved factoriality without restriction on the dimension, but it cannot be easily generalized to the general commutation relation case. We generalize the result of Śniady and present a simpler proof. Our estimate for the number of generators in case q > 0 is better than in [Snia].
4
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Measures connected with Bargmann's representation of the q-commutation relation for q > 1

100%
Królak I.
Banach Center Publications
|
1998
|
tom 43
|
nr 1
253-257
EN
Classical Bargmann's representation is given by operators acting on the space of holomorphic functions with scalar product $〈z^n,z^k〉_q = δ_{n,k}[n]_q! = F(z^n \bar{z}^k)$. 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.
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