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Strong σ-closedness of σ-subspaces

Determine whether every σ-subspace of a σ-C*-algebra is strongly σ-closed; in particular, show whether the image of a σ-injection A → B is strongly σ-closed in B.

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Background

The paper introduces strong σ-closure (closure under order limits) and shows that kernels of σ-normal maps are strongly σ-closed. However, it is not known whether arbitrary σ-subspaces enjoy this property, which affects stability properties of σ-substructures under embeddings.

References

It remains unclear whether σ-subspaces are strongly σ-closed in general. For example, given a σ-injection $A\hookrightarrow B$ of s, we do not know whether the image of $A$ is strongly σ-closed in $B$.

— Categories of abstract and noncommutative measurable spaces (2504.13708 - Fritz et al., 18 Apr 2025) in Interlude on strong σ-closures (Section on σ-completions)