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 $\O_{\theta}$of a 2-graph $\Fth$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of $\O_{\theta}$ and its unitary pairs with a \textit{twisted property}. We characterize when endomorphisms preserve the fixed point algebra $\fF$ of the gauge automorphisms and its canonical masa $\fD$. Some other properties of endomorphisms are also investigated. As far as the modular theory of $\O_{\theta}$ is concerned, we show that the algebraic *-algebra generated by the generators of $\O_{\theta}$ with the inner product induced from a distinguished state $\omega$ is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra $\pi(\O_{\theta})"$ generated by the GNS representation of $\omega$ is an AFD factor of type III$_1$, provided $\frac{\ln m}{\ln n}\not\in\bQ$. Here $m,n$ are the numbers of generators of $\Fth$ of degree $(1,0)$ and $(0,1)$, respectively. This work is a continuation of \cite{DPY1, DPY2} by Davidson-Power-Yang and \cite{DY} by Davidson-Yang.