TY - JOUR

T1 - Commutants of weighted shift directed graph operator algebras

AU - Levene, Rupert

AU - Kribs, David

AU - Power, Stephen Charles

N1 - First published in Proceedings of the American Mathematical Society in 145, 2017, published by the American Mathematical Society

PY - 2017

Y1 - 2017

N2 - We consider non-selfadjoint operator algebras $\Lalg\lambda$ 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 $\Lalg{\lambda}$ in the case of the single vertex graph with two edges and a suitable choice of left weight function~$\lambda$.

AB - We consider non-selfadjoint operator algebras $\Lalg\lambda$ 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 $\Lalg{\lambda}$ in the case of the single vertex graph with two edges and a suitable choice of left weight function~$\lambda$.

KW - Directed graph

KW - weighted shift

KW - non-selfadjoint algebra

KW - commutant

KW - left regular representation

KW - creation operators

KW - Fock space

U2 - 10.1090/proc/13477

DO - 10.1090/proc/13477

M3 - Journal article

VL - 145

SP - 3465

EP - 3480

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

ER -