Quasicentral approximate identity - Imported Knowlpedia Content

Definition

Let $I$ be a closed two-sided ideal in a $C^*$ -algebra $A$ . A quasicentral approximate identity for $I$ in $A$ is a positive contractive approximate identity $(e_\lambda)$ for $I$ such that

\lVert e_\lambda a-ae_\lambda\rVert\longrightarrow0 \qquad(a\in A).

Thus the net must approximate the identity on $I$ from both sides and must asymptotically commute, in norm, with the entire ambient algebra. The phrase “in $A$ ” matters: the same approximate identity can be quasicentral relative to one containing algebra and not another.

Existence theorem

Every closed two-sided ideal $I\triangleleft A$ has a quasicentral approximate identity in $A$ . Starting from an ordinary approximate identity for $I$ , one uses convex combinations to make finitely many commutators small while preserving the approximation properties; directing the construction by finite subsets and error tolerances yields the required net Arveson, §1.7.

The theorem guarantees a net, not necessarily a sequence. Countability hypotheses can allow sequential forms, but the definition itself should not replace the indexing net by natural numbers.

Examples and a near-miss

If $A$ is commutative, every approximate identity for every ideal is automatically quasicentral. If $A$ is unital and $I=A$ , the constant net $e_\lambda=1_A$ is quasicentral.

In contrast, let $P_n$ project $\ell^2(\mathbb N)$ onto the first $n$ standard basis vectors. The sequence $(P_n)$ is an approximate identity for $K(\ell^2)$ , but it is not quasicentral in $B(\ell^2)$ : for the unilateral shift $S$ , one has $\lVert P_nS-SP_n\rVert=1$ . This does not contradict the existence theorem, which may require a different net.

Extension-theoretic role

Quasicentral approximate identities let computations in an ideal approach central behavior relative to an extension algebra. They are used to separate ideal and quotient contributions, construct asymptotic splittings, and control commutators in extension theory. Quasicentrality does not mean that any $e_\lambda$ is central, nor does it imply norm convergence of $(e_\lambda)$ to an element of $A$ Blackadar, §II.4.

References

William Arveson, An Invitation to $C^*$ -Algebras, Graduate Texts in Mathematics 39, Springer, 1976. DOI record. Relevant: §1.7 on quasicentral approximate units.
Bruce Blackadar, Operator Algebras: Theory of $C^*$ -Algebras and von Neumann Algebras, Springer, 2006. DOI record. Relevant: §II.4 on approximate units and quasicentrality.