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We introduce a class of generalized universal irrational rotation$C^*$-algebras $A_{\theta,\gamma}=C^*(x,w)$ which is characterized by therelations $w^*w=ww^*=1$, $x^*x=\gamma(w)$, $xx^*=\gamma(e^{-2\pi i\theta}w)$,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 traciallinear functionals, simplicity, and $K$-groups of $A_{\theta,\gamma}$ in termsof zero points of $\gamma(z)$. We show that if $A_{\theta,\gamma}$ is simplethen $A_{\theta,\gamma}$ is an $A{\mathbb T}$-algebra of real rank zero. Weclassify $A_{\theta,\gamma}$ in terms of $\theta$ and zero points of$\gamma(z)$. Let $A_\theta=C^*(u,v)$ be the universal irrational rotation$C^*$-algebra with $vu=e^{2\pi i\theta}uv$.  Then $C^*(u+v)\cong A_{\theta,|1+z|^2}$. As an application, we show that$C^*(u+v)$ is a proper simple $C^*$-subalgebra of $A_\theta$ which has a uniquetrace, $K_1(C^*(u+v))\cong \mathbb{Z}$, and there is an order isomorphism of$K_0(C^*(u+v))$ onto $\mathbb{Z}+\mathbb{Z}\theta$. {Moreover, $C^*(u+v)$ is aunital simple $A{\mathbb T}$-algebra of real rank zero.} We also calculate thespectrum and the Brown measure of $u+v$.

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