Unital C*-algebra - Knowlpedia

A $C^*$ -algebra $A$ is unital if there is an element $1_A\in A$ such that

1_Aa=a1_A=a\qquad(a\in A).

The identity is unique. Except in the zero algebra, it is self-adjoint, positive, and has norm one. Unitality is part of the ambient algebraic structure, not a choice of an approximate identity. A $*$ -homomorphism between unital $C^*$ -algebras need not preserve their identities unless it is explicitly required to be unital.

Consequences of having a unit

In a unital $C^*$ -algebra, invertibility and the spectrum of $a$ are defined internally using $1_A$ . The continuous functional calculus for a normal element is unital, and scalar multiples $\lambda1_A$ provide a canonical copy of $\mathbb C$ when $A\neq0$ . A closed ideal of a unital algebra may still be nonunital; it has a unit precisely when it is generated by a central projection.

Unitization

Every nonunital $C^*$ -algebra $A$ admits a unitization $\widetilde A$ containing $A$ as a codimension-one closed ideal and fitting into

0\longrightarrow A\longrightarrow\widetilde A\longrightarrow\mathbb C \longrightarrow0.

This construction lets one discuss spectra, invertibility, and unitary elements without pretending that the adjoined identity lies in $A$ . Different concrete embeddings may supply larger units, so the canonical unitization and a multiplier algebra should not be conflated.

Examples and conventions

The algebra $C(X)$ is unital for compact Hausdorff $X$ , with constant function $1$ , whereas $C_0(X)$ is unital exactly when $X$ is compact. The compact operators on an infinite-dimensional Hilbert space are nonunital. Some authors regard the zero algebra as unital with $1=0$ , while others require $1\neq0$ ; statements using $\|1_A\|=1$ implicitly exclude the zero case.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on units, unitizations, spectra, and morphism conventions.