Nondegenerate representation of a C*-algebra - Imported Knowlpedia Content

Definition

Let $\pi:A\to\mathcal B(H)$ be a representation of a $C^*$ -algebra. It is nondegenerate if

\overline{\operatorname{span}\{\pi(a)\xi:a\in A,\ \xi\in H\}}=H.

Equivalently, no nonzero vector is annihilated by every operator $\pi(a)$ . This is the representation-space instance of a nondegenerate $*$ -homomorphism. It does not mean that $\pi$ is injective: faithfulness controls the kernel in $A$ , whereas nondegeneracy says that the represented algebra has no zero orthogonal summand in $H$ .

Approximate-identity criteria

If $(e_i)$ is an approximate identity of $A$ , then $\pi$ is nondegenerate exactly when

\pi(e_i)\xi\longrightarrow\xi\qquad(\xi\in H).

Thus $\pi(e_i)$ converges strongly to $I_H$ , independently of the chosen approximate identity. If $A$ is unital, the condition reduces to $\pi(1_A)=I_H$ . These equivalences are proved in Pedersen, the chapter on representations.

Essential subspace

For an arbitrary representation, the essential subspace

H_{\mathrm{ess}}=\overline{\operatorname{span}\pi(A)H}

reduces $\pi$ . The restriction to $H_{\mathrm{ess}}$ is nondegenerate, while the restriction to $H_{\mathrm{ess}}^\perp$ is zero. Hence every representation decomposes canonically as a nondegenerate representation plus a zero representation.

Extension to multipliers

A nondegenerate representation extends uniquely to a unital representation of the multiplier algebra $M(A)$ , with convergence in the strict topology carried to convergence in the strong operator topology. This extension property is a principal reason that nondegenerate representations are the standard convention for nonunital $C^*$ -algebras.

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 3 on representations, approximate identities, and nondegeneracy.
Dana P. Williams, Crossed Products of C-Algebras*, Mathematical Surveys and Monographs 134, American Mathematical Society, 2007. DOI record. Relevant: Appendix A on nondegenerate homomorphisms and multiplier extensions.