Bounded Maps Between Close Operator Algebras

Definitions and mathematical statement

For normed algebras $A$ and $B$ over a common scalar field, a linear map $T\colon A\to B$ is bounded if \[ \|T\|:=\sup_{\substack{a\in A\\ \|a\|\leq 1}}\|T(a)\|<\infty. \] Equivalently, there exists $C\geq 0$ such that $\|T(a)\|\leq C\|a\|$ for every $a\in A$.

The supplied source asks: What methods produce bounded linear maps between close, or completely close, operator algebras?

This is a methodological question with unresolved formulation details. The supplied record does not specify the class of operator algebras, their representations, the meaning or quantitative threshold of either closeness condition, or the additional properties required of the constructed maps. In particular, the zero map $T\colon A\to B$, defined by $T(a)=0$ for every $a\in A$, is always bounded and linear. Thus existence of a bounded linear map alone does not express a substantive perturbation problem. Any intended nontriviality, approximation, invertibility, multiplicativity, or quantitative norm requirement must be recovered from the source context before a precise theorem-like target can be stated. No such requirement is imposed in this draft.

Short English statement

The source asks for methods constructing bounded linear maps between operator algebras that are close or completely close. It leaves the closeness conditions and the desired properties of the maps unspecified; these are essential because the zero map already satisfies boundedness and linearity.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002393$, source problem code catalogue entry, title: Bounded Maps Between Close Operator Algebras. The record lists proposed year 2013 and identifies no individual proposer.
• Supplied report citation: C*-Algebren (2014), Oberwolfach Reports, pp. 2423--2500, DOI 10.4171/owr/2013/43. Original OWR report: https://doi.org/10.4171/owr/2013/43. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled "Christensen et al. primary paper": https://doi.org/10.1007/s11511-012-0075-5. No paper title, full author list, or specific result is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes bounded-map constructions as known in special perturbation settings, and describes the general methodological problem as remaining open. This is supplied dated curation, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement exact and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. This verification claim is supplied provenance and has not been independently checked for this draft.