Generation of Tail Algebras by Diagonal Subalgebras

Definitions and mathematical statement

For a von Neumann algebra $M$ and a subset $S\subseteq M$, define the von Neumann algebra generated by $S$ inside $M$, using the unital convention, by \[ W_M^*(S)=\bigcap\{N\subseteq M:N\text{ is a von Neumann subalgebra},\ 1_N=1_M,\ S\subseteq N\}. \] The supplied question is: is the tail algebra $T_\phi$ generated as a von Neumann algebra by $D_\phi$? Provided that the intended objects are subalgebras of a common ambient von Neumann algebra $M$, with the intended common-unit convention, this asks whether \[ T_\phi=W_M^*(D_\phi). \] The supplied record does not define $\phi$, its admissible class, the underlying algebra or representation, the tail algebra $T_\phi$, or $D_\phi$. It also does not state the quantifier over $\phi$. Consequently, the displayed equality is a schematic rendering of the question, not a complete quantified formulation. The title refers to diagonal subalgebras, but supplies no definition identifying $D_\phi$. These missing definitions and conventions must be recovered from the cited source before a precise standalone statement can be finalized.

Short English statement

The question asks whether the specified subalgebra $D_\phi$ generates the entire tail von Neumann algebra $T_\phi$. The supplied record leaves the objects, hypotheses, and quantification over $\phi$ undefined, so the formulation remains provisional.

Sources

• Local UnsolvedMath registry, supplied record for permanent ID $30002897$, source problem code catalogue entry, titled "Generation of Tail Algebras by Diagonal Subalgebras," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2015.
• Supplied source citation: Free Probability Theory (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/28, pp. 1571--1629. https://doi.org/10.4171/owr/2015/28. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: "Oberwolfach report 28." https://doi.org/10.4171/owr/2015/28.
• Supplied literature source: "Quantum symmetric states on free product C-star algebras." https://arxiv.org/abs/1305.7293.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the later report still lists generation of the von Neumann tail algebra $T$ by $D$ as open, and that related de Finetti and amalgamated-freeness theorems do not establish this generation statement. This is supplied dated curation, not an independently verified literature assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.