Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case

Ilona Królak. "Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case." Banach Center Publications 73.1 (2006): 277-284. <http://eudml.org/doc/281825>.

@article{IlonaKrólak2006,
abstract = {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].},
author = {Ilona Królak},
journal = {Banach Center Publications},
keywords = {general commutation relations; von Neumann algebras; factors},
language = {eng},
number = {1},
pages = {277-284},
title = {Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case},
url = {http://eudml.org/doc/281825},
volume = {73},
year = {2006},
}

TY - JOUR
AU - Ilona Królak
TI - Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case
JO - Banach Center Publications
PY - 2006
VL - 73
IS - 1
SP - 277
EP - 284
AB - 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].
LA - eng
KW - general commutation relations; von Neumann algebras; factors
UR - http://eudml.org/doc/281825
ER -