KMS-Symmetric Quantum Markov Semigroups on von Neumann Algebras

Definitions and mathematical statement

Let $M$ be a von Neumann algebra. In the usual unital convention, a quantum Markov semigroup is a family $(T_t)_{t\geq 0}$ of normal, unital, completely positive maps $T_t:M\to M$ such that \[ T_0=\operatorname{id}_M,\qquad T_{s+t}=T_sT_t\quad(s,t\geq0), \] with the appropriate continuity in $t$. The supplied excerpt does not specify that continuity convention. If the semigroup is continuous in operator norm, its bounded generator on $M$, with one common sign convention, is \[ A(x)=\lim_{t\downarrow0}\frac{T_t(x)-x}{t}. \] The supplied record does not establish whether its phrase "bounded generators" refers to generators on $M$ or to their implementations on an associated $L^2$-space, nor does it specify the sign convention.

KMS symmetry requires reference data, ordinarily a faithful normal state or a suitable faithful normal semifinite weight. For example, when a faithful normal state $\varphi$ is specified, write $j_\varphi(x)=\varphi^{1/4}x\varphi^{1/4}$ for the symmetric embedding into the associated noncommutative $L^2$-space. In this convention, KMS symmetry means \[ \langle j_\varphi(T_t(x)),j_\varphi(y)\rangle = \langle j_\varphi(x),j_\varphi(T_t(y))\rangle \quad(x,y\in M,\ t\geq0). \] This state-based description is conditional notation, not an additional hypothesis attributed to the source: an arbitrary von Neumann algebra need not admit a faithful normal state, and the intended state or weight framework is absent from the supplied excerpt.

An $M$-correspondence is a Hilbert space $\mathcal H$ with commuting normal unital representations of $M$ and its opposite algebra, giving left and right actions. Given a suitable algebra $\mathcal A\subseteq M$, a derivation into $\mathcal H$ is a linear map $\delta:\mathcal A\to\mathcal H$ satisfying \[ \delta(xy)=x\cdot\delta(y)+\delta(x)\cdot y \quad(x,y\in\mathcal A). \] These definitions alone do not specify the derivation characterization mentioned in the record.

Requested extension, with unresolved qualifications. Determine whether the necessary-and-sufficient derivation/correspondence characterization of KMS-symmetric quantum Markov semigroups asserted in the source report for matrix algebras extends to arbitrary von Neumann algebras, with the reconstructed registry statement restricting attention to bounded generators. The original excerpt conjectures that the matrix-algebra characterization continues to hold in this generality. However, it omits the characterization itself, including the relation between the generator and the derivation, the admissible correspondences, the derivation domain, and any modular or reality conditions. Consequently, the supplied evidence does not determine a complete quantified equivalence. An explicit characterization theorem cannot be stated precisely without that missing material.

Short English statement

The supplied conjecture asks whether a derivation-based characterization of KMS-symmetric quantum Markov semigroups on matrix algebras remains valid for arbitrary von Neumann algebras. The registry specifies bounded generators, but the excerpt omits the characterization and its reference-state or weight assumptions. This is therefore a qualified statement-only draft requiring the missing theorem and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006565$, source problem code catalogue entry, titled "KMS-Symmetric Quantum Markov Semigroups on von Neumann Algebras." No individual proposer is identified. The record labels the problem open and the reconstructed statement corrected_verified; these are supplied curation labels, not independent verification here.
• Noncommutative Harmonic Analysis and Quantum Information (2026), Oberwolfach Reports, pp. 217--275, DOI https://doi.org/10.4171/owr/2026/4. The supplied original excerpt carries the report page number 225 and refers to previous joint work with Vernooij, labelled [VW23]. It asserts a full characterization on matrix algebras and conjectures its extension to arbitrary von Neumann algebras, without reproducing the characterization.
• Supplied literature source: "Oberwolfach Report 4/2026," https://doi.org/10.4171/owr/2026/4.
• Supplied literature source: "Vernooij--Wirth, Derivations and KMS-Symmetric QMS," https://doi.org/10.1007/s00220-023-04795-6. The record supplies no theorem statement from this reference.
• Literature assessment dated exactly 2026-08-22: the supplied curation states that the derivation representation is established for matrix algebras, that substantial results for general von Neumann algebras are known, and that the report's full characterization conjecture remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.