Title: Endomorphisms and modular theory of 2-graph C*-algebras

Authors: Dilian Yang

Issue: Volume 59 (2010), Issue 2, 495-520

Abstract: In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras $\mathcal{O}_{\theta}$ of a 2-graph $\mathbb{F}_{\theta}^{+}$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of $\mathcal{O}_{\theta}$ and its unitary pairs with a \textit{twisted property}. We study when endomorphisms preserve the fixed point algebra $\mathfrak{F}$ of the gauge automorphisms and its canonical masa $\mathfrak{D}$. Some other properties of endomorphisms are also investigated.\par As far as the modular theory of $\mathcal{O}_{\theta}$ is concerned, we show that the algebraic *-algebra generated by the generators of $\mathcal{O}_{\theta}$ with the inner product induced from a distinguished state $\omega$ is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra $\pi(\mathcal{O}_{\theta})''$ generated by the GNS representation of $\omega$ is an AFD factor of type III$_1$, provided $\ln m/\ln n \not\in \mathbb{Q}$. Here $m$, $n$ are the numbers of generators of $\mathbb{F}_{\theta}^{+}$ of degree $(1,0)$ and $(0,1)$, respectively.\par This work is a continuation of [Davidson, K.R., Power, S.C., Yang, D., \textit{Atomic representations of rank 2 graph algebras}, J. Funct. Anal. \textbf{255} (2008), 819--853; Davidson, K.R., Power, S.C., Yang, D., \textit{Dilation theory for rank 2 graph algebras}, J. Operator Theory (to appear); Davidson, K.R., Yang, D., \textit{Periodicity in rank 2 graph algebras}, Canad. J. Math. \textbf{61} (2009), 1239--1261].