The Bass and topological stable ranks for algebras of almost periodic functions on the real line (Q2790582)

In the \(K\)-theory of operator algebras, the concept of topological stable rank of a \(C^\ast\)-algebra, introduced and studied by Marc Rieffel more than thirty years ago, is of central importance. This rank dominates the Bass stable rank (well studied in algebraic \(K\)-theory) and the two coincide for commutative \(C^\ast\)-algebras, as Rieffel observed. The paper under review is concerned with \(C^\ast\)-algebras of almost periodic functions on the real line. For the algebra \(AP\) of all almost periodic functions on \(\mathbb{R}\), Suárez showed two decades ago that both ranks are infinite. Extending this result, it is shown in the present paper that these ranks are infinite also for algebras of the form \(AP_{\Lambda}:= \{f\in AP: \sigma(f) \subset \Lambda\}\) when \(\Lambda\) is a subsemigroup of \(\mathbb{R}\) such that the \(\mathbb{Q}\)-linear span of \(\Lambda\) is infinite dimensional, \(\sigma(f)\) denoting the Bohr spectrum of \(f\). The conclusion fails without the dimension assumption on \(\Lambda\).