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

Banach Center Publications (2006)

- Volume: 73, Issue: 1, page 277-284
- ISSN: 0137-6934

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topIlona 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 -

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