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

Ilona Królak

Ilona Królak

Banach Center Publications, Tome 72 (2006), p. 277-284 / Harvested from The Polish Digital Mathematics Library

Résumé

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: $a_{i} a ⁺_{j} = \sum_{r, s} t_{j}^{i} r_{s} a ⁺_{r} a_{s} + δ_{i j} I d$ . 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 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].

Publié le : 2006-01-01

Zbl 1103.81025

EUDML-ID : urn:eudml:doc:281825

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     author = {Ilona Kr\'olak},
     title = {Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case},
     journal = {Banach Center Publications},
     volume = {72},
     year = {2006},
     pages = {277-284},
     zbl = {1103.81025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc73-0-20}
}

Ilona Królak. Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case. Banach Center Publications, Tome 72 (2006) pp. 277-284. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc73-0-20/