Endomorphisms and Modular Theory of 2-graph C*-algebras

In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras Oθ of a 2-graph F + θ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of Oθ and its unitary pairs with a twisted property. We characterize when endomorphisms preserve the fixed point algebra F of the gauge automorphisms and its canonical masa D. Some other properties of endomorphisms are also investigated. As far as the modular theory of Oθ is concerned, we show that the algebraic *-algebra generated by the generators of Oθ with the inner product induced from a distinguished state ω is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra π(Oθ) ′′ generated by the GNS representation of ω is an AFD factor of type III1, provided lnm lnn 6∈ Q. Here m,n are the numbers of generators of F θ of degree (1, 0) and (0, 1), respectively. This work is a continuation of [11, 12] by Davidson-Power-Yang and [13] by Davidson-Yang.