## Advanced Studies in Pure Mathematics

### Extensions of quasidiagonal $C^*$-algebras and K-theory

#### Abstract

Let $0 \to I \to E \to B \to 0$ be a short exact sequence of $C^*$-algebras where $E$ is separable, $I$ is quasidiagonal (QD) and $B$ is nuclear, QD and satisfies the UCT. It is shown that if the boundary map $\partial : K_1(B) \to K_0(I)$ vanishes then $E$ must be QD also.

A Hahn-Banach type property for $K_0$ of QD $C^*$-algebras is also formulated. It is shown that every nuclear QD $C^*$-algebra has this $K_0$-Hahn-Banach property if and only if the boundary map $\partial : K_1(B) \to K_0(I)$ (from above) always completely determines when $E$ is QD in the nuclear case.

#### Article information

Dates
First available in Project Euclid: 1 January 2019

https://projecteuclid.org/ euclid.aspm/1546368825

Digital Object Identifier
doi:10.2969/aspm/03810065

Mathematical Reviews number (MathSciNet)
MR2059801

Zentralblatt MATH identifier
1065.46034

#### Citation

Brown, Nathanial P.; Dadarlat, Marius. Extensions of quasidiagonal $C^*$-algebras and K-theory. Operator Algebras and Applications, 65--84, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/03810065. https://projecteuclid.org/euclid.aspm/1546368825