$C^*$-algebra extensions associated to continued fraction expansions of rational numbers
Jack Spielberg
TL;DR
The paper solves the isomorphism problem for essential unital extensions $0 \to \mathcal{K} \oplus \mathcal{K} \to E \xrightarrow{\pi} M_n \otimes C(\mathbb{T}) \to 0$ by introducing the index $\mathrm{ind}(\tau)=(a_+,a_-) \in \mathbb{Z}^2$ and a robust invariant $\overline{m}$ in $\mathbb{Z}^2/(\mathbb{Z} a + n \mathbb{Z}^2)$, where $a=(a_+,a_-)^T$ is the index; the isomorphism class of $E$ is determined up to the action of $(-1)\cdot$ and coordinate exchange by the pair (index orbit, $\overline{m}$). Specializing to the rational-case analogs of Effros-Shen algebras, the authors construct C*-algebras from categories of paths and groupoids associated to finite- nonzero sequences $k=(k_i)$, obtaining a one-to-one correspondence between such sequences and isomorphism classes of extensions with index $(-1,1)$; in particular, there is a bijection between $[0,1)\cap \mathbb{Q}$ and these extension classes with $\overline{m}$ coprime to $n$. They further relate these rational-analytic objects to the continued-fraction data via a nonsimple continued-fraction encoding $[0,1,k_1,1,k_2,1,\ldots]$, and outline a future construction of an upper semicontinuous field over $[0,1)$. Overall, the work provides a complete invariant and a concrete rational-classification scheme for a natural family of C*-algebra extensions, linking operator algebra invariants to number-theoretic continued-fraction data.
Abstract
We solve the isomorphism problem for essential unital $C^*$-algebra extensions of the form $0 \to \mathcal{K} \oplus \mathcal{K} \to E \xrightarrowπ M_n \otimes C(\mathbb{T}) \to 0$. We then relate these to analogs of the Effros Shen AF algebras for rational numbers. This involves $C^*$-algebras constructed from categories of paths built from certain nonsimple continued fraction expansions.