Quotient C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and let $I$ be a closed two-sided ideal of $A$ . The quotient $C^*$ -algebra $A/I$ is the algebraic quotient with

(a+I)^*=a^*+I \qquad\text{and}\qquad \lVert a+I\rVert=\inf_{x\in I}\lVert a+x\rVert .

These operations are well defined, the quotient is complete, and the $C^*$ -identity holds. The canonical map $q:A\to A/I$ , $q(a)=a+I$ , is a surjective $*$ -homomorphism with kernel $I$ . Closedness of $I$ is essential: a quotient by a nonclosed ideal does not carry this Hausdorff $C^*$ -norm.

Universal property and exactness

If $\varphi:A\to B$ is a $*$ -homomorphism satisfying $I\subseteq\ker\varphi$ , there is a unique $*$ -homomorphism $\overline{\varphi}:A/I\to B$ such that $\varphi=\overline{\varphi}\circ q$ . Thus quotient maps are precisely the surjective morphisms in the usual category of $C^*$ -algebras, up to isomorphism of the codomain. The sequence

0\longrightarrow I\longrightarrow A\overset{q}{\longrightarrow}A/I \longrightarrow 0

is the basic short exact sequence associated with $I$ Murphy, §3.1.

Units and representative examples

When $A$ is unital and $I$ is proper, $A/I$ is unital with identity $1_A+I$ . A quotient of a nonunital algebra can nevertheless be unital. For $A=C(X)$ and a closed subset $F\subseteq X$ , the ideal of functions vanishing on $F$ has quotient canonically isomorphic to $C(F)$ . For bounded operators on a Hilbert space, the quotient by the compact operators is the Calkin algebra.

Conventions and boundary cases

Some authors allow the zero $C^*$ -algebra to be unital and others require $0\ne1$ . This affects only the description of $A/A$ , not its quotient operations. The closedness hypothesis belongs to the definition in the $C^*$ -setting even when “ideal” elsewhere means an algebraic two-sided ideal. The quotient norm is determined by the ideal; it is not an independently chosen completion norm.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §3.1 on ideals, quotient norms, and quotient $C^*$ -algebras.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 1 on ideals and quotients.