Mathematics – Operator Algebras
Scientific paper
2009-07-07
Mathematics
Operator Algebras
Some changed were made for Proposition 4.2 (i). To appear in IUMJ
Scientific paper
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.
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