  We investigate Cuntz-Pimsner $C^*$-algebras associated with certain
correspondences of the unit circle $\mathbb{T}$. We analyze these
$C^*$-algebras by analogy with irrational rotation algebras $A_\theta$ and
Cuntz algebras $\mathcal{O}_n$. We construct a Rieffel type projection, study
the fixed point algebras of certain actions of finite groups, and calculate the
entropy of a certain endomorphism. We also study the induced map of the dual
action of the gauge action on $K$-groups.
