Canad. Math. Bull. 46(2003), 388-399
Printed: Sep 2003
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 $0\to J\to E\to A\to 0$ 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.
tracially quasidiagonal extensions, tracial rank
46L05 - General theory of $C^*$-algebras
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]