Free Polar Decomposition Under a Nondegeneracy Condition

Definitions and mathematical statement

Write $E\colon\mathcal A\to B$ for the $B$-valued expectation implicit in the source record, where $z\in\mathcal A$. In the usual operator-valued probability terminology, such an expectation fixes $B$, is unital, and satisfies $E(b_1ab_2)=b_1E(a)b_2$ for $a\in\mathcal A$ and $b_1,b_2\in B$. A $B$-valued circular element is conventionally characterized by the vanishing of all operator-valued cumulants in $z,z^*$, with intervening coefficients from $B$, except possibly second-order cumulants containing one $z$ and one $z^*$. The supplied excerpt does not specify the ambient operator-algebraic assumptions or the precise source conventions for the decomposition terms below.

Define the moment span \[ M_z:=\operatorname{span}\left\{E((z^*z)^k):k\in\mathbb Z_{\geq0}\right\}\subseteq B, \] where the zeroth power is the identity. Let $\tau$ denote the topology intended by the source for the density condition; that topology is not identified in the supplied record.

Question. In the setting intended by the cited report, is it true that every $B$-valued circular element $z$ which admits a free bipolar decomposition and satisfies \[ \overline{M_z}^{\,\tau}=B \] admits a free polar decomposition?

Unresolved qualifications. The record does not define either a free bipolar decomposition or a free polar decomposition. In particular, it does not specify the factors permitted in these decompositions, the expectation or state relative to which freeness is required, whether freeness is amalgamated over $B$, or whether a decomposition may be realized in an enlarged probability space. These conventions, the ambient assumptions on $\mathcal A,B,E$, and the topology $\tau$ must be recovered from the cited source before this question becomes a fully specified mathematical statement. No choice among them is asserted here.

Short English statement

The question asks whether a $B$-valued circular element with a free bipolar decomposition must also have a free polar decomposition when the linear span of its expected positive moments is dense in $B$. The supplied excerpt leaves the decomposition conventions, ambient setting, and density topology unspecified, so this remains a qualified statement-only draft.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30005925$, title Free Polar Decomposition Under a Nondegeneracy Condition, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The proposed year is recorded as 2024; no individual proposer is supplied.
• Non-commutative Function Theory and Free Probability (2024), Oberwolfach Reports, pp. 1227--1284, DOI 10.4171/owr/2024/22. Supplied report reference and literature-source link: https://doi.org/10.4171/owr/2024/22. The background describes this report as documenting workshop discussions and open problems.
• Dykema--Griffin, supplied literature-source link: https://arxiv.org/abs/2306.17133. The record supplies no paper title or precise supporting location.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that degenerate examples and partial structural theorems are known, while existence under the stated dense-span nondegeneracy condition remains explicitly open. This is supplied dated curation, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft. The retained original excerpt explicitly states the dense-span condition but does not reproduce the full decomposition question.