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

Abstract

In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras ${Ø}_{\theta }$of a 2-graph $\Fth$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of ${Ø}_{\theta }$ 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 ${Ø}_{\theta }$ is concerned, we show that the algebraic *-algebra generated by the generators of ${Ø}_{\theta }$ with the inner product induced from a distinguished state $\omega$ is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra $\pi ({Ø}_{\theta })"$ 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.

Some changed were made for Proposition 4.2 (i). To appear in IUMJ

References in corpus (2)

• Simplicity of C*-algebras associated to higher-rank graphs
• Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras