Short exact sequence of C*-algebras

A short exact sequence of $C^*$ -algebras is a diagram

0\longrightarrow I\xrightarrow{\iota}A\xrightarrow{q}B \longrightarrow 0

of $*$ -homomorphisms in which $\iota$ is injective, $q$ is surjective, and $\operatorname{im}\iota=\ker q$ . After identifying $I$ with its image, $I$ is a closed two-sided ideal of $A$ , and the induced map from $A/I$ to $B$ is an isometric $*$ -isomorphism. The sequence is also called an extension of $B$ by $I$ ; this word order records that $I$ is the ideal and $B$ is the quotient.

Canonical ideal–quotient sequence

Every closed two-sided ideal $I\triangleleft A$ gives the exact sequence

0\longrightarrow I\longrightarrow A\longrightarrow A/I \longrightarrow 0.

Conversely, every short exact sequence of $C^*$ -algebras is isomorphic to one of this form. The closedness of $I$ is automatic when it is the kernel of a $*$ -homomorphism, and it is essential for $A/I$ to carry its quotient $C^*$ -norm.

Splittings

The extension is split if there is a $*$ -homomorphism $s:B\to A$ with $q\circ s=\operatorname{id}_B$ . A bounded linear or a completely positive section is weaker and does not make the extension split as a $C^*$ -algebra extension. Even when a $*$ -splitting exists, the middle algebra need not be a direct product: the section can encode a nontrivial action of $B$ on $I$ .

Functorial significance

The underlying vector-space or module sequence is a short exact sequence, but the $C^*$ -structure supplies extra analytic rigidity: injective $*$ -homomorphisms are isometric, quotient norms are fixed, and ideals are closed. A functor on $C^*$ -algebras is called exact when it preserves these sequences. Operator $K$ -theory instead produces a cyclic six-term exact sequence from them.

Multiplier example

For any $C^*$ -algebra $I$ , its embedding as an essential ideal in the multiplier algebra yields

0\longrightarrow I\longrightarrow M(I) \longrightarrow M(I)/I\longrightarrow 0.

The quotient $M(I)/I$ is the corona algebra. Extensions with ideal $I$ can often be encoded by homomorphisms from the quotient algebra into this corona algebra.

References

Bruce Blackadar, Operator Algebras: Theory of C-Algebras and von Neumann Algebras*, Springer, 2006. DOI record. Relevant: §§II.5 and II.8 on ideals, quotients, and extensions.
Niels E. Wegge-Olsen, K-Theory and C-Algebras: A Friendly Approach*, Oxford University Press, 1993. DOI record. Relevant: Chapters 2–3 on multiplier algebras and extensions.