Commutator Models for Nonsymmetric Free Random Variables

Definitions and mathematical statement

A Borel probability measure $\mu$ on ${\mathbb{R}}$ is symmetric about zero if $\mu(B)=\mu(-B)$ for every Borel set $B\subseteq{\mathbb{R}}$, where $-B=\{-x:x\in B\}$.

To explain the free-probability terminology, let $(\mathcal A,\varphi)$ be a unital $*$-probability space: $\varphi$ is a positive linear functional with $\varphi(1)=1$. Unital $*$-subalgebras $\mathcal A_1,\mathcal A_2$ are free if \[ \varphi(x_1\cdots x_n)=0 \] for every $n\geq1$ and every choice $x_j\in\mathcal A_{\epsilon_j}$ with $\varphi(x_j)=0$, $\epsilon_j\in\{1,2\}$, and $\epsilon_j\neq\epsilon_{j+1}$. Elements $a,b$ are free when the unital $*$-subalgebras they generate are free. In a self-adjoint operator setting where spectral distributions exist, the law $\mu_a$ is characterized by $\varphi(f(a))=\int_{{\mathbb{R}}}f\,{\,\mathrm{d}}\mu_a$ for the functions admitted by the functional calculus, and similarly for $b$.

The algebraic commutator is \[ [a,b]=ab-ba. \] For self-adjoint $a,b$, whenever the products and adjoints are defined as required, the normalized commutator $c=i(ab-ba)$ is self-adjoint. The supplied record does not specify whether its representations concern $[a,b]$, this normalized commutator, or another convention.

Requested extension, with unresolved scope. Extend the existing representations describing commutators of free random variables with symmetric marginal laws $\mu_a,\mu_b$ to arbitrary marginal probability laws, without requiring either marginal to be symmetric. The intended extension should apply to every pair of free variables in the admissible class of the original representations and recover those representations when both marginal laws are symmetric.

The supplied record does not identify the representations to be extended or state their hypotheses. Consequently, the preceding request is a statement of the intended research objective, not a fully specified formula or proposition. The definitions above explain a usual self-adjoint interpretation; the record does not establish that this is the complete intended setting. In particular, no choice of boundedness, support, moment, trace, or operator-domain assumptions is asserted here.

Short English statement

The problem asks for an extension of commutator representations for free random variables from symmetric marginal distributions to distributions with no symmetry assumption. The supplied record leaves the representations, commutator convention, and admissible class unspecified, so a fully precise target requires source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004022$, source problem code catalogue entry, entitled Commutator Models for Nonsymmetric Free Random Variables; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied.
• Supplied report citation: Free Probability Theory (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/53, pp. 3147--3215. Supplied source URL and literature reference labeled "Oberwolfach report 53": https://doi.org/10.4171/owr/2018/53. The registry separately records the proposed year as 2018.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, states that the report explicitly identifies the general nonsymmetric extension as open, and reports that no authoritative primary source located through its cutoff removes that generality. This is supplied dated curation, not an independently verified assessment; the report and its cited claims have not been independently checked for this draft.
• The registry labels the statement corrected_verified and describes its changes as preserving mathematical content while repairing presentation or definitions. This supplied editorial label does not establish mathematical correctness or resolve the formulation issues recorded here.