Finite Blaschke Symbols and the $K$-Theory of Scalar Odometer $C^*$-Algebras

Let $O_n$ be the odometer semigroup, and let $\mathcal A_\xi=C^*(S_1,\ldots,S_n,W_\xi)\subseteq\mathcal B(\mathcal F_n^2)$ be the $C^*$-algebra generated by the left creation operators and the scalar odometer map associated with a symbol $\xi\in\mathcal F_n^2$. We show that $\mathcal A_\xi$ contains the compact operators for every scalar symbol. For an isometric scalar symbol, we prove that $W_\xi$ is Fredholm if and only if the associated inner function is a finite Blaschke product. We further show that the image of $\mathcal A_\xi$ in the Calkin algebra is canonically isomorphic to the odometer boundary quotient $\mathcal Q(O_n)$.
If the associated finite Blaschke product has degree $d$, then $\operatorname{ind}(W_\xi)=-d$. For $d\geq 1$, we obtain $K_0(\mathcal A_\xi)\cong\mathbb Z\oplus\mathbb Z_{d(n-1)}$ and $K_1(\mathcal A_\xi)=0$, whereas for $d=0$, $K_0(\mathcal A_\xi)\cong\mathbb Z^2$ and $K_1(\mathcal A_\xi)\cong\mathbb Z$. Consequently, for fixed $n\geq 2$, finite Blaschke symbols of distinct degrees generate non-isomorphic $C^*$-algebras.