Endomorphisms and Modular Theory of 2-Graph C*-Algebras

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In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras Øθof a 2-graph on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of Øθ and its unitary pairs with a twisted property. We characterize when endomorphisms preserve the fixed point algebra of the gauge automorphisms and its canonical masa. Some other properties of endomorphisms are also investigated. As far as the modular theory of Øθ is concerned, we show that the algebraic *-algebra generated by the generators of Øθ with the inner product induced from a distinguished state ω is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra π(Øθ)" generated by the GNS representation of ω is an AFD factor of type III₁, provided m/ n∈. Here m,n are the numbers of generators of of degree (1,0) and (0,1), respectively. This work is a continuation of DPY1, DPY2 by Davidson-Power-Yang and DY by Davidson-Yang.