Unitization of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a nonunital $C^*$ -algebra. Its unitization $\widetilde A$ is the vector space $A\oplus\mathbb C$ with

(a,\lambda)(b,\mu)=(ab+\lambda b+\mu a,\lambda\mu), \qquad (a,\lambda)^*=(a^*,\overline\lambda).

It has the canonical $C^*$ -norm extending the norm of $A$ , identity $(0,1)$ , and an isometric embedding $a\mapsto(a,0)$ . Under this embedding, $A$ is a closed essential ideal of the unital $C^*$ -algebra $\widetilde A$ , and the scalar map $(a,\lambda)\mapsto\lambda$ identifies $\widetilde A/A$ with $\mathbb C$ .

Universal extension property

For every $*$ -homomorphism $\varphi:A\to B$ into a unital $C^*$ -algebra, there is a unique unital $*$ -homomorphism

\widetilde\varphi:\widetilde A\to B, \qquad \widetilde\varphi(a,\lambda)=\varphi(a)+\lambda1_B,

extending $\varphi$ . This property characterizes the unitization up to canonical unital $*$ -isomorphism. It also records why the algebraic construction must be equipped with its $C^*$ -norm and involution rather than treated as merely adjoining a ring identity Murphy, §2.1.

Spectra, characters, and multipliers

For $a\in A$ , the spectrum used in nonunital $C^*$ -algebra theory is computed from $(a,0)$ in $\widetilde A$ ; zero necessarily belongs to this spectrum. The scalar quotient is a character $\epsilon:\widetilde A\to\mathbb C$ . The unitization embeds canonically in the multiplier algebra $M(A)$ , but it can be strictly smaller: for example, the unitization of the compact operators is $\mathcal K(H)+\mathbb C I_H$ , whereas their multiplier algebra is $B(H)$ .

Conventions for already unital algebras

When $A$ already has an identity, some authors define its unitization to be $A$ itself. Others retain the formula $A\oplus\mathbb C$ , which adjoins a new identity and is then isomorphic to $A\oplus\mathbb C$ as a unital algebra by a change of coordinates. The core definition assumes $A$ is nonunital, so these conventions do not conflict. “Minimal” distinguishes this construction from the generally larger multiplier algebra.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §2.1 on adjoining an identity and nonunital spectra.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.3 on unitization and its canonical $C^*$ -norm.