Commutants of weighted shift directed graph operator algebras

TL;DR

This paper characterizes the commutants of weighted shift operator algebras associated with directed graphs, revealing conditions for the double commutant property and providing examples where it fails.

Contribution

It provides a complete description of the commutant for these algebras and identifies conditions under which the double commutant property holds or fails.

Findings

Complete description of the commutant of $rak{L}(G,\lambda)$

Conditions ensuring the double commutant property

Counterexample where the double commutant property fails

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 $\lambda$.