Applications of CPH Semigroups and Dilations

Definitions and mathematical statement

Terminology and unresolved scope. The supplied record names CPH-maps, CPH-semigroups, and CPH-dilations but supplies no mathematical definitions of these structures. In particular, their underlying spaces, admissible maps, compatibility conditions, semigroup parameter sets, and dilation requirements are unspecified. Their definitions must be established from the cited sources before a precise mathematical formulation can be completed.

Research request as supplied. Determine useful applications of CPH-maps, CPH-semigroups, and CPH-dilations, including whether these structures yield significant new dilation or operator-theoretic results.

The record gives no specific application, proposed theorem, class of operators, or quantified dilation assertion to establish. The words "useful" and "significant new" have no supplied mathematical criteria. Consequently, this is an application-seeking research request with unresolved definitions and targets, rather than a fully specified conjecture. No additional hypotheses or quantifiers are imposed in this draft.

Short English statement

Identify applications of CPH-maps, CPH-semigroups, and CPH-dilations, particularly applications yielding dilation or operator-theoretic results. The supplied record does not define these structures or specify a theorem to prove, so a precise mathematical target remains unresolved.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003808$, title Applications of CPH Semigroups and Dilations, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The recorded proposed year is 2018; no individual proposer is supplied.
• Supplied report citation: Interactions between Operator Space Theory and Quantum Probability with Applications to Quantum Information (2019), Oberwolfach Reports, pp. 1293--1379, DOI 10.4171/owr/2018/22. https://doi.org/10.4171/owr/2018/22. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or passage is supplied.
• Supplied literature reference: CPH extendable maps and semigroups. https://doi.org/10.1016/j.jmaa.2014.01.024.
• Supplied literature reference: Oberwolfach Report 22. https://doi.org/10.4171/owr/2018/22.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the CPH framework and semigroups are defined and developed, while the extracted request for concrete useful applications or a general dilation role remains unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.