Nondegenerate *-homomorphism - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be $C^*$ -algebras. A $*$ -homomorphism $\varphi:A\to M(B)$ is nondegenerate if

\overline{\operatorname{span}}\{\varphi(a)b:a\in A,\ b\in B\}=B.

Here $M(B)$ acts canonically on $B$ . A representation $\pi:A\to B(H)$ is nondegenerate when $\overline{\pi(A)H}=H$ . These conditions exclude a nonzero summand on which the image acts as zero and remain meaningful when $A$ or $B$ is nonunital. Nondegeneracy is the morphism hypothesis that permits canonical extensions to multiplier algebras.

Approximate-identity criteria

If $(e_i)$ is an approximate identity for $A$ , then $\varphi$ is nondegenerate exactly when $\varphi(e_i)b\to b$ for every $b\in B$ . Equivalently, $\varphi(e_i)\to1_{M(B)}$ strictly. For a representation, the corresponding criterion is strong convergence $\pi(e_i)\xi\to\xi$ for every $\xi\in H$ . These criteria are independent of the chosen approximate identity Pedersen, §3.12.

Multiplier extension

Every nondegenerate $\varphi:A\to M(B)$ has a unique unital strictly continuous extension

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

that agrees with $\varphi$ on $A$ . Conversely, the restriction of such an extension is nondegenerate. This is why nondegenerate homomorphisms, rather than arbitrary $*$ -homomorphisms, are used for multiplier-algebra functoriality, correspondences, and covariant representations Lance, §2.

Unital and degenerate cases

If $A$ is unital, nondegeneracy is equivalent to $\varphi(1_A)=1_{M(B)}$ . A representation is degenerate precisely when the orthogonal complement of $\overline{\pi(A)H}$ is nonzero; restricting to the essential subspace $\overline{\pi(A)H}$ produces a nondegenerate representation. A zero map into a nonzero target is degenerate, while a map into the zero algebra is sensitive to the convention adopted for its identity.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. DOI record. Relevant: §2 on nondegenerate homomorphisms and multiplier extensions.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.12 on approximate identities and nondegenerate representations.
Dana P. Williams, Crossed Products of C-Algebras*, American Mathematical Society, 2007. DOI record. Relevant: Appendix A on multiplier algebras and nondegenerate homomorphisms.