Approximate identity in a C*-algebra

Let $A$ be a $C^*$ -algebra. An approximate identity is a net $(e_\lambda)$ in $A$ such that, for every $a\in A$ ,

\lim_\lambda e_\lambda a=a \quad\text{and}\quad \lim_\lambda a e_\lambda=a

in norm. A contractive approximate identity additionally satisfies $\lVert e_\lambda\rVert\leq 1$ ; a positive contractive approximate identity has positive terms with $0\leq e_\lambda\leq 1$ . Every $C^*$ -algebra has a positive contractive approximate identity, so that form is usually meant by “approximate unit” in $C^*$ -algebra theory.

Existence and construction

One construction directs the positive contractions of $A$ by how well they act as identities on finite subsets. Functional calculus applied to finite sums of elements $a^*a$ then produces a net that works simultaneously on both sides. For a closed two-sided ideal $I$ , the same theorem supplies an approximate identity lying inside $I$ . In a unital algebra the constant net $e_\lambda=1_A$ is the simplest example.

Sequential and strictly positive cases

An approximate identity need not be a sequence. A $C^*$ -algebra admits a countable approximate identity precisely when it is $\sigma$ -unital. If $h\in A$ is strictly positive, suitable continuous functions of $h$ , such as $h(h+1/n)^{-1}$ in the unitization, form a sequential approximate identity. Separability implies $\sigma$ -unitality, but many nonseparable $C^*$ -algebras require genuinely nonsequential nets.

Multipliers and nondegeneracy

Inside the multiplier algebra $M(A)$ , every approximate identity converges strictly to the multiplier unit: $e_\lambda a\to a$ and $ae_\lambda\to a$ for all $a\in A$ . Likewise, a representation $\rho:A\to B(\mathcal H)$ is nondegenerate exactly when $\rho(e_\lambda)\xi\to\xi$ for every $\xi\in\mathcal H$ . These facts make approximate identities the correct replacement for an absent unit, while not asserting norm convergence to a unit outside $A$ .

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §3.1 on ideals and approximate identities.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.4 on approximate units.