Endomorphisms of certain irrational rotation \(C^*\)-algebras (Q1203585)

Let \(\theta\) be a quadratic irrational number with its discriminant \(D=5\). Let \(A_ \theta\) be the corresponding irrational rotation \(C^*\)-algebra. In the present paper we construct a unital endomorphism \(\Phi\) of \(A_ \theta\) with \(\Phi_ *\) an arbitrary endomorphism of \(K_ 1(A_ \theta)\). Furthermore we show that there is a unital endomorphism \(\varphi\) of \(A_ \theta\) with the minimizing index \([A_ \theta\): \(\varphi(A_ \theta)]_ 0=4\).