Nonself-adjoint 2-graph algebras

Fuller, Adam; Yang, Dilian

doi:10.1090/S0002-9947-2015-06385-5

Nonself-adjoint $2$-graph algebras
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by Adam H. Fuller and Dilian Yang PDF

Trans. Amer. Math. Soc. 367 (2015), 6199-6224 Request permission

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.

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