Spectral Subspaces of Measures on Infinite Groups

Definitions and mathematical statement

Extracted question. For an infinite group $G$ and $\mu\in M_0(G)\setminus\{0\}$, is\[N_{|\mu|}^{(p)}=L_1^{(p)}(m_\mu)?\] The intended quantifiers are over the permitted infinite groups, nonzero measures in the source's class $M_0(G)$, and the permitted values of $p$, but neither that group class nor the parameter range is supplied. All four source-specific constructions $M_0(G)$, $N_{|\mu|}^{(p)}$, $m_\mu$, and $L_1^{(p)}(m_\mu)$ remain undefined in the record. In an ordinary complex-measure setting $|\mu|$ denotes total variation, but even the measure-theoretic hypotheses must be recovered before that convention identifies the displayed objects.

A precise statement requires the topology and structural assumptions on $G$, the class and scalar field of measures, the definition of $M_0(G)$, the admissible $p$, the construction and codomain of $m_\mu$, and the ambient space or identification in which the proposed equality is asserted. Equality of sets, equality of normed spaces with equivalent norms, and isometric equality are different possible conclusions. The notation $L_1^{(p)}$ is not silently replaced by an ordinary scalar $L^p$ space, nor is $M_0(G)$ assigned a guessed Fourier-decay or zero-mass definition. Thus the exact symbolic question is retained with a substantive formulation gap.

The supplied literature triage dated 2026-08-21 labels the item open but discusses a Hilbert-space case and a converse characterization of Hilbert spaces, citing work on vector-valued Bloch functions. The extracted question gives no Banach-space parameter or Bloch-function construction connecting it to that assessment. This apparent mismatch must be reconciled before treating the triage as evidence of any progress on the displayed equality. The triage's statement that no later general resolution was found is a supplied dated search report, not an independent status claim.

Short English statement

The record asks whether two source-defined spaces associated with a nonzero measure on an infinite group coincide. Their definitions and the range of $p$ are absent. The attached Hilbert-space and Bloch-function assessment is not connected to the displayed equality by the supplied text.

Sources

• Local UnsolvedMath registry, permanent ID 30000101, code catalogue entry; Spectral Subspaces of Measures on Infinite Groups, Oberwolfach Reports Open Problems.
• Spectral Theory in Banach Spaces and Harmonic Analysis (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/36, pp. 1883--1970. https://doi.org/10.4171/owr/2004/36. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label Blasco, vector-valued Bloch functions: https://www.uv.es/oblasco/Investigacion/BST/Bloch.pdf; its relevance to this equality is not established by the record.
• Supplied literature label original Oberwolfach report: https://doi.org/10.4171/owr/2004/36.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified label concerns removal of a repeated label and balanced math notation, not recovery of the missing definitions.