Advanced Studies in Pure Mathematics

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

Nathanial P. Brown and Marius Dadarlat

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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

Operator Algebras and Applications, H. Kosaki, ed. (Tokyo: Mathematical Society of Japan, 2004), 65-84

First available in Project Euclid: 1 January 2019

Permanent link to this document euclid.aspm/1546368825

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]


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.

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