σ-unital C*-algebra - Imported Knowlpedia Content

Definition

A $C^*$ -algebra $A$ is $\sigma$ -unital if it has a sequential approximate identity: there are positive contractions $e_1,e_2,\ldots\in A$ such that

\lVert e_n a-a\rVert\longrightarrow0 \quad\text{and}\quad \lVert a e_n-a\rVert\longrightarrow0

for every $a\in A$ . Equivalently, $A$ has a countable approximate identity; a finite approximate identity may be repeated to form a sequence. The prefix $\sigma$ expresses this countability condition and does not refer to a topology on $A$ .

Equivalent characterizations

A $C^*$ -algebra is $\sigma$ -unital exactly when it contains a strictly positive element $h$ . In that case, continuous functional calculus gives a canonical sequential approximate identity

e_n=f_n(h),\qquad f_n(t)=\min\{nt,1\}.

Conversely, a rapidly weighted sum of the terms of a countable approximate identity produces a strictly positive element. These equivalences are standard consequences of approximate-unit theory Pedersen, §1.4.

Examples and permanence

Every unital $C^*$ -algebra is $\sigma$ -unital, using the constant sequence $1_A$ . Every separable $C^*$ -algebra is $\sigma$ -unital, although the converse fails: an unital algebra can be nonseparable. For a locally compact Hausdorff space $X$ , $C_0(X)$ is $\sigma$ -unital exactly when $X$ is $\sigma$ -compact. If $H$ is a separable Hilbert space, the finite-rank projections onto an increasing sequence of finite-dimensional subspaces form an approximate identity for $K(H)$ .

The property need not pass to arbitrary ideals. If $D$ is an uncountable discrete space and $D^+$ its one-point compactification, then $C(D^+)$ is unital, but its ideal $C_0(D)$ is not $\sigma$ -unital. In contrast, a hereditary subalgebra generated by one positive element is automatically $\sigma$ -unital.

Why countability matters

Countable approximate identities allow arguments with sequences instead of general nets. They are a standing hypothesis in much of multiplier-algebra, Hilbert-module, and $C^*$ -classification theory. The condition is weaker than separability and stronger than merely possessing an approximate identity, since every $C^*$ -algebra has the latter Lance, Chapter 2.

References

Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.4 on countable approximate units and strictly positive elements.
E. Christopher Lance, Hilbert $C^*$ -Modules: A Toolkit for Operator Algebraists, Cambridge University Press, 1995. DOI record. Relevant: Chapter 2 on approximate units and sigma-unital $C^*$ -algebras.