The tracial topological rank of extensions of $C^*$-algebras

Let $0\to \mathcal J\to \mathcal A\to \mathcal A / \mathcal J\to 0$ be a short exact sequence of separable $C^*$-algebras. We introduce the notion of tracially quasidiagonal extension. Suppose that $\mathcal J$ and $\mathcal A/J$ have tracial topological rank zero. We prove that if $(\mathcal A, \mathcal J)$ is tracially quasidiagonal, then $\mathcal A$ has tracial topological rank zero.

Tracially Quasidiagonal Extensions

It is known that a unital simple C*-algebra A with tracial topological rank zero has real rank zero. We show in this note that, in general, there are unital C*-algebras with tracial topological rank zero that have real rank other than zero.Let 0 → J → E → A → 0 be a short exact sequence of C*-algebras. Suppose that J and A have tracial topological rank zero. It is known that E has tracial topological rank zero as a C*-algebra if and only if E is tracially quasidiagonal as an extension. We present an example of a tracially quasidiagonal extension which is not quasidiagonal.