Multiplier algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. Its multiplier algebra $M(A)$ is the unital $C^*$ -algebra of double centralizers of $A$ . An element is a compatible pair $(L,R)$ of bounded linear maps $A\to A$ satisfying

aL(b)=R(a)b\qquad(a,b\in A).

Every $a\in A$ defines a multiplier by left and right multiplication, giving a canonical injective $*$ -homomorphism

A\longrightarrow M(A),\qquad a\longmapsto(L_a,R_a).

Under this embedding, $A$ is an essential ideal of $M(A)$ .

Universal role

If a $C^*$ -algebra $B$ contains $A$ as an essential closed two-sided ideal, multiplication of elements of $B$ on $A$ gives a canonical injective $*$ -homomorphism $B\to M(A)$ that restricts to the standard embedding of $A$ . This is the precise sense in which $M(A)$ is the largest unital $C^*$ -algebra containing $A$ essentially. The qualifier “essential” matters: without it, one could adjoin unrelated direct summands.

Fundamental examples

If $A$ is unital, then every multiplier comes from an element of $A$ , so $M(A)=A$ . For a locally compact Hausdorff space $X$ ,

M(C_0(X))\cong C_b(X),

the bounded continuous functions, whereas the unitization generally gives only functions on the one-point compactification. If $H$ is a Hilbert space and $\mathcal K(H)$ is its compact-operator $C^*$ -algebra, then

M(\mathcal K(H))\cong\mathcal B(H),

so bounded operators are precisely its multipliers.

Nondegenerate morphisms

A nondegenerate $*$ -homomorphism $\varphi:A\to M(B)$ extends uniquely to a unital $*$ -homomorphism

\overline{\varphi}:M(A)\to M(B)

that is continuous for the strict topologies. This extension principle is why multiplier-valued morphisms, rather than only maps $A\to B$ , are natural in crossed products, Hilbert $C^*$ -modules, and nonunital noncommutative geometry.

Strict versus norm approximation

An approximate identity $(e_i)$ of $A$ converges to $1_{M(A)}$ in the strict topology, meaning $e_ia\to a$ and $ae_i\to a$ in norm for every $a\in A$ . It usually does not converge in the norm of $M(A)$ . Thus $M(A)$ is not generally the norm completion of $A$ ; it adds multipliers visible through their left and right actions.

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.12 on double centralizers and multiplier algebras.
E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. DOI record. Relevant: Chapter 2 on multiplier algebras and nondegenerate homomorphisms.
Robert C. Busby, “Double Centralizers and Extensions of C-Algebras,” Transactions of the American Mathematical Society* 132 (1968), 79–99. DOI record.