Finite dimensional representations of the soft torus (Q2759021)

Let \(\varepsilon\geq 0\) and \(A_\varepsilon\) denote the universal (unital) \(C^\ast\)-algebra generated by a pair of unitaries \(u,v\) subject to the relation \(\|uv-vu\|\leq \varepsilon\). This class of algebras, referred to as soft tori, naturally interpolates \(C(T^2)\) (\(\varepsilon\geq 2\)) and the algebra \(F_2\) (\(\varepsilon\geq 2\)). It is proved that \(A_\varepsilon\) is residually finite dimensional (RFD) in the sense that it possess a separating family of finite dimensional representations. The proof is based on showing that \(A_\varepsilon\) is a cross product of an auxiliary RFD algebra by the group \(Z\). As an immediate corollary it is shown that \(A_\varepsilon\) has a faithful tracial state and that any hyponormal operator in \(A_\varepsilon\) is normal.