Corona C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra, canonically embedded as an essential ideal in its multiplier algebra $M(A)$ . The corona algebra of $A$ is the quotient $C^*$ -algebra

Q(A)=M(A)/A.

Its elements are multiplier classes modulo those implemented by elements of $A$ . The definition applies to every $C^*$ -algebra, but it is most useful for nonunital $A$ : if $A$ is unital, then $M(A)=A$ and $Q(A)=0$ . Some authors write $\mathcal Q(A)$ or $\mathcal C(A)$ instead of $Q(A)$ .

Fundamental examples

For an infinite-dimensional Hilbert space $H$ ,

Q(\mathcal K(H))\cong\mathcal B(H)/\mathcal K(H),

the Calkin algebra, because $M(\mathcal K(H))\cong\mathcal B(H)$ , with $\mathcal K(H)$ the compact operators and $\mathcal B(H)$ the bounded operators. In the commutative case $A=C_0(X)$ , where $X$ is locally compact Hausdorff, $M(A)\cong C_b(X)$ ; after identifying $C_b(X)$ with $C(\beta X)$ , the corona is $C(\beta X\setminus X)$ . The topological remainder motivates the name “corona.”

Extension-theoretic role

An extension

0\longrightarrow A\longrightarrow E\longrightarrow B\longrightarrow0

whose copy of $A$ is essential determines a $*$ -homomorphism $\tau:B\to Q(A)$ , called its Busby invariant. Conversely, a suitable $*$ -homomorphism into $Q(A)$ reconstructs an extension by a pullback. This correspondence is the reason corona algebras organize extension theory Busby, §§3–4.

Scope and cautions

The corona is not the unitization of $A$ : it is a quotient of the generally much larger multiplier algebra. Nor must $Q(A)$ be nonzero when $A\ne0$ , as the unital case shows. Structural properties such as simplicity, separability, and exactness do not pass automatically from $A$ to its corona.

References

Robert C. Busby, “Double Centralizers and Extensions of C-Algebras,” Transactions of the American Mathematical Society* 132 (1968), 79–99. AMS DOI record. Relevant: §§3–4 on multiplier quotients, Busby invariants, and extensions.
E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Cambridge DOI record. Relevant: Chapter 2 on multiplier algebras and their quotients.