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

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