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arXiv · 1605.03758

Commutants of weighted shift directed graph operator algebras

David W. Kribs·Rupert H. Levene·Stephen C. Power
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Abstract

We consider non-selfadjoint operator algebras $\mathfrak{L}(G,\lambda)$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative 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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David W. Kribs, Rupert H. Levene, Stephen C. Power. 2016-05-12. Commutants of weighted shift directed graph operator algebras. https://doi.org/10.1090/proc/13477

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