On generalized universal irrational rotation algebras and the operator $u+v$

Abstract

We introduce a class of generalized universal irrational rotation $C^{\ast }$-algebras $A_{\theta ,\gamma }=C^{\ast }(x,w)$ which is characterized by the relations $w^{\ast }w=w{w}^{\ast }=1$, $x^{\ast }x=\gamma (w)$, $x{x}^{\ast }=\gamma (e^{-2\pi i\theta }w)$, and $xw=e^{-2\pi i\theta }wx$, where $\theta$ is an irrational number and $\gamma (z)\in C(\mathbb{T})$ is a positive function. We characterize tracial linear functionals, simplicity, and $K$-groups of $A_{\theta ,\gamma }$ in terms of zero points of $\gamma (z)$. We show that if $A_{\theta ,\gamma }$ is simple then $A_{\theta ,\gamma }$ is an $A\mathbb{T}$-algebra of real rank zero. We classify $A_{\theta ,\gamma }$ in terms of $\theta$ and zero points of $\gamma (z)$. Let $A_{\theta }=C^{\ast }(u,v)$ be the universal irrational rotation $C^{\ast }$-algebra with $vu=e^{2\pi i\theta }uv$. Then $C^{\ast }(u+v)\cong A_{\theta ,|1+z{|}^2}$. As an application, we show that $C^{\ast }(u+v)$ is a proper simple $C^{\ast }$-subalgebra of $A_{\theta }$ which has a unique trace, $K_1(C^{\ast }(u+v))\cong \mathbb{Z}$, and there is an order isomorphism of $K_0(C^{\ast }(u+v))$ onto $\mathbb{Z}+\mathbb{Z}\theta$. {Moreover, $C^{\ast }(u+v)$ is a unital simple $A\mathbb{T}$-algebra of real rank zero.} We also calculate the spectrum and the Brown measure of $u+v$.

51 pages