Extension of a nondegenerate homomorphism to multiplier algebras

Statement

Let $A$ and $B$ be $C^*$ -algebras. Every nondegenerate $*$ -homomorphism $\varphi:A\to M(B)$ into the multiplier algebra of $B$ has a unique unital $*$ -homomorphism

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

whose restriction to $A\subseteq M(A)$ is $\varphi$ . The extension is continuous from the strict topology on $M(A)$ to the strict topology on $M(B)$ . It is characterized by

\overline{\varphi}(m)\varphi(a)b=\varphi(ma)b

for $m\in M(A)$ , $a\in A$ , and $b\in B$ . Thus nondegeneracy is precisely the hypothesis that makes the extension canonical and unital.

Construction and uniqueness

The dense subspace $\operatorname{span}\varphi(A)B$ of $B$ determines the left action of a multiplier $m$ by

\overline{\varphi}(m)\bigl(\varphi(a)b\bigr)=\varphi(ma)b.

The corresponding right-multiplier relation makes this formula independent of the chosen expression and yields an element of $M(B)$ . Density gives uniqueness. Equivalently, for any approximate identity $(e_i)$ of $A$ ,

\overline{\varphi}(m)=\operatorname*{strict\,lim}_i\varphi(me_i).

This is the extension theorem in Lance, Proposition 2.1.

Functorial consequences

If $\psi:B\to M(C)$ is nondegenerate, the composite used in the nonunital category is $\overline{\psi}\circ\varphi:A\to M(C)$ , and its multiplier extension is $\overline{\psi}\circ\overline{\varphi}$ . A nondegenerate representation $\pi:A\to B(H)=M(\mathcal K(H))$ therefore extends uniquely to $M(A)$ . These formulas are fundamental in covariant representations and crossed products Williams, Appendix A.

Scope and near-miss

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. DOI record. Relevant: Proposition 2.1 on extension of nondegenerate homomorphisms.
Dana P. Williams, Crossed Products of C-Algebras*, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Appendix A on multiplier algebras and nondegenerate homomorphisms.