Nonself-adjoint 2-graph algebras

TL;DR

This paper investigates the structure of weakly-closed nonself-adjoint algebras generated by 2-graph representations, revealing a lower-triangular matrix form and conditions for von Neumann algebra properties.

Contribution

It introduces a new structural decomposition for 2-graph algebras and characterizes when these algebras are von Neumann algebras.

Findings

Algebras have a lower-triangular 3x3 form.

Conditions identified for algebras to be von Neumann.

Analysis of atomic representations included.

Abstract

We study the structure of weakly-closed nonself-adjoint algebras arising from representations of single vertex 2-graphs. These are the algebras generated by 2 isometric tuples which satisfy a certain commutation relation. We show that these algebras have a lower-triangular $3\times 3$ form. The left-hand side of this matrix decomposition is a slice of the enveloping von Neumann algebra generated by the 2-graph algebra. We further give necessary and sufficient conditions for these algebras themselves to be von Neumann algebras. The paper concludes with further study of atomic representations.