Commutants of weighted shift directed graph operator algebras

We consider non-selfadjoint operator algebras L(G,λ)\mathcal {L} (G,\lambda ) generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs GG. These algebras may be viewed as non-commutative generalizations of weighted Bergman space algebras or as weighted version...

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Veröffentlicht in:Proceedings of the American Mathematical Society 2017-08, Vol.145 (8), p.3465-3480
Hauptverfasser: Kribs, David W., Levene, Rupert H., Power, Stephen C.
Format: Artikel
Sprache:eng
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Zusammenfassung:We consider non-selfadjoint operator algebras L(G,λ)\mathcal {L} (G,\lambda ) generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs GG. These algebras may be viewed as non-commutative generalizations of weighted Bergman space algebras or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for L(G,λ)\mathfrak {L} (G,\lambda ) in the case of the single vertex graph with two edges and a suitable choice of left weight function λ\lambda.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/13477