Von Neumann Generation of the Algebra T-Psi

Definitions and mathematical statement

Let $A$ be a unital $C^*$-algebra, and let $A_j$, $j\geq 1$, denote copies of $A$. Write \[ B=*_{j=1}^{\infty}A_j. \] Here the intended convention is the full unital free product with the units identified; the supplied record does not explicitly specify this convention. For a permutation $\sigma$ of ${\mathbb{N}}$ fixing all but finitely many indices, let $\alpha_\sigma$ permute the copies of $A$ accordingly. A state $\psi$ on $B$ is symmetric if \[ \psi\circ\alpha_\sigma=\psi \qquad\text{for every such }\sigma. \] Let $(\pi_\psi,H_\psi,\xi_\psi)$ be its GNS representation, and put \[ M_\psi=W^*(\pi_\psi(B)),\qquad T_\psi=\bigcap_{n\geq1}W^*\!\left(\bigcup_{j\geq n}\pi_\psi(A_j)\right). \] For a subset of $\mathcal B(H_\psi)$, $W^*$ denotes the von Neumann algebra it generates in this representation. The vector state extending $\psi$ to $M_\psi$ is $\widetilde\psi(x)=\langle x\xi_\psi,\xi_\psi\rangle$.

The source designates a $C^*$-subalgebra $D_\psi\subseteq T_\psi$ as the associated tail $C^*$-algebra, obtained from a canonical state-preserving conditional expectation onto $T_\psi$. State preservation means $\widetilde\psi(E_\psi(x))=\widetilde\psi(x)$ on the domain of that expectation. The supplied record does not specify that domain or the construction of $D_\psi$.

The intended question is whether, for every unital $C^*$-algebra $A$ and every symmetric state $\psi$ in this setup, the source-designated algebra satisfies \[ W^*(D_\psi)=T_\psi. \] The target equality is explicit, but its fully precise formulation requires the missing definition of $D_\psi$ and the hypotheses for the stated conditional expectation. No particular construction of $D_\psi$ is asserted here. The supplied literature assessment identifies the unresolved scope as the general symmetric case outside the quantum symmetric case; this is a supplied qualification, not an independently verified result.

Short English statement

For a symmetric state on an infinite free product, ask whether the associated tail $C^*$-algebra $D_\psi$ generates the entire tail von Neumann algebra $T_\psi$ in the GNS representation. The supplied record leaves the construction of $D_\psi$ incomplete. Its open-status label and qualification concerning the quantum symmetric case require independent review.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30003208$, source problem code catalogue entry, title "Von Neumann Generation of the Algebra T-Psi," in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2016 and no individual proposer.
• Supplied report citation: $C^*$-Algebras (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/40, pp. 2269--2345. https://doi.org/10.4171/owr/2016/40. The displayed report year is retained as supplied; its relationship to the DOI year and proposed year requires verification.
• Supplied literature reference: Dykema, Kostler and Williams, "Quantum symmetric states." https://arxiv.org/abs/1305.7293.
• Supplied literature reference: "OWR Free Probability Theory report." https://ems.press/content/serial-article-files/46575.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the primary treatment explicitly asks this question outside the quantum symmetric case, and reports that no general resolution was found. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the setup. These are supplied verification claims; no source visit or independent verification is asserted in this draft.