Finite Blaschke Symbols and the $K$-Theory of Scalar Odometer $C^{\ast }$-Algebras

Abstract

Let $O_n$ be the odometer semigroup, and let ${\mathcal{A}}_{\xi }=C^{\ast }(S_1,\dots ,S_n,W_{\xi })\subseteq \mathcal{B}({\mathcal{F}}_n^2)$ be the $C^{\ast }$-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 $\mathrm{ind}(W_{\xi })=-d$. For $d\ge 1$, 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 $n\ge 2$, finite Blaschke symbols of distinct degrees generate non-isomorphic $C^{\ast }$-algebras.

14 pages