Commutants of weighted shift directed graph operator algebras

Kribs, David; Levene, Rupert; Power, Stephen

doi:10.1090/proc/13477

Commutants of weighted shift directed graph operator algebras
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by David W. Kribs, Rupert H. Levene and Stephen C. Power PDF

Proc. Amer. Math. Soc. 145 (2017), 3465-3480 Request permission

Abstract:

We consider non-selfadjoint operator algebras $\mathcal {L} (G,\lambda )$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. 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 $\mathfrak {L} (G,\lambda )$ in the case of the single vertex graph with two edges and a suitable choice of left weight function $\lambda$.

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