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Minghui Ma

Publications and source records attributed to Minghui Ma.

At least 19 recordsLinked to original sources
arXiv2026

The ultrapower $\mathcal R^ω$ is not a group factor

Let $(\mathcal M_n)_{n\geqslant 1}$ be a sequence of type $\mathrm{II}_1$ factors with property $Γ$ and let $ω$ be a free ultrafilter on $\mathbb N$. We prove that every subfactor of the ultraproduct $\prod_ω\mathcal M_n$ of finite Jones index is not isomorphic to the group von Neumann algebra of any discrete group. In particular, the ultrapower $\mathcal R^ω$ of the hyperfinite type $\mathrm{II}_1$ factor $\mathcal R$ is not a group factor.

math.OA↗
arXiv2026

Single commutators in $C^*$-algebras

Let $\mathcal A$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $\mathcal T(\mathcal A)$, and strict comparison of projections. We prove that every element $T$ in $\mathcal A$ with $τ(T)=0$ for all $τ\in\mathcal T(\mathcal A)$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal A$ has unique trace; $(2)$ $\mathcal A$ has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^*$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.

math.OA↗
arXiv2026

Strongly irreducible operators, $K_0$-groups, and Jones indices in the hyperfinite type $\mathrm{II}_1$ factor

In this paper, we study the strong irreducibility of a class of operators associated with irrational rotation algebras in the hyperfinite type $\mathrm{II}_1$ factor $\mathcal R$. By developing weighted shift models for their relative commutants, we obtain $K_0$ calculations and splitting criteria for direct sums. In particular, we construct strongly irreducible operators in $\mathcal R$ whose relative commutants have $K_0$-groups isomorphic to $\mathbb Z$ and $\mathbb Z^2$. We realize every Jones index of an irreducible subfactor of $\mathcal R$ as $[\mathcal R:W^*(T)]$ for some strongly irreducible operator $T$ relative to $\mathcal R$. Furthermore, we construct strong generators of $\mathcal R$ and strongly irreducible operators in every type $\mathrm{II}_1$ factor $\mathcal M$ with separable predual.

math.OA↗
arXiv2026

Proactive Dialogue Policy Optimization via Cognitive-State Transition

Proactive dialogue requires agents to continually adapt their policies to user feedback while progressing toward task objectives over multiple turns. To move beyond imitation learning on static datasets, recent approaches use user simulators to collect interactive data for policy optimization. However, many simulators do not explicitly model the evolution of user cognition, limiting the consistency and state dependence of feedback across turns. Moreover, representing each action only by a high-level strategy label overlooks the large utterance space and cannot distinguish alternative realizations of the same strategy. To this end, we jointly design a $\textbf{Cog}$nitive User $\textbf{Sim}$ulator $\textbf{(Cog-Sim)}$ and $\textbf{C}$ognitive-$\textbf{S}$tate $\textbf{T}$ransition--Driven $\textbf{P}$olicy $\textbf{O}$ptimization $\textbf{(CSTPO)}$. Cog-Sim maintains the user's cognitive and affective states and generates responses through constrained state transitions across turns, so feedback depends on both the realized utterance and the user's current state. CSTPO organizes each action as a hierarchical strategy--utterance representation: a high-level strategy label constrains utterance sampling, and utterances are optimized within each label. Sparse complete-branch sampling reuses shared dialogue prefixes and estimates separate strategy-level and utterance-level advantages, enabling fine-grained optimization at both levels. Across three tasks, Cog-Sim exhibits monotonic dose--response relationships and is preferred over prompt-based simulators for naturalness. CSTPO improves Qwen3-14B's performance to a level comparable to that of GPT-5.5-based planning methods.

cs.AI↗
arXiv2026

Cognitive World Model for Progressive BDI/E Trajectory Evaluation of Conversational Agents

As LLM-based conversational agents advance toward increasingly open-ended and interaction-intensive scenarios, task completion alone provides an incomplete assessment of their effectiveness. The evolution of users' internal states, including beliefs, desires, intentions, and emotions (BDI/E), serves as an intermediate signal connecting agent behaviors with interaction outcomes and reflects how conversational strategies shape users during multi-turn interactions. However, existing evaluation paradigms primarily focus on surface-level responses or final outcomes, providing limited insight into the underlying cognitive processes. This limitation makes it difficult to diagnose why agents succeed or fail and to optimize their interaction strategies. To address this challenge, we propose Cognitive World Model (CogWM), an LLM-based cognitive user model that jointly models users' BDI/E states and corresponding responses, enabling explicit cognitive trajectory tracking. Trained on 150K user-turn samples with Qwen3-14B, CogWM achieves superior performance over existing user simulation baselines in both response fidelity and cognitive state understanding. Interactions with six state-of-the-art LLMs demonstrate that CogWM enables progressive comparison of agents through cognitive trajectories, revealing distinct agent patterns and complementary relationships between cognitive evolution and behavioral outcomes.

cs.AI↗
arXiv2026

Kaplansky's second test problem in operator algebras

Kaplansky's second test problem on similarity asks: if $T$ and $S$ are elements in a unital Banach algebra $\mathcal{B}$ and $T\oplus T$ is similar to $S\oplus S$ in $\mathbb{M}_2(\mathcal{B})$, is $T$ similar to $S$ in $\mathcal{B}$? We answer this problem affirmatively if $T$ is an operator with property $(J)$ in a type $\mathrm{I}_n$ von Neumann algebra $\mathcal{M}$, i.e., $\{T\}'\cap\mathcal{M}$ contains a bounded maximal abelian family of idempotents. Moreover, the condition of property $(J)$ can be removed for $1\leqslant n\leqslant 3$. A similar result is proved if $T$ is an element in a unital Banach algebra $\mathcal{B}$ with essentially finite-dimensional commutant, i.e., the relative commutant of $T$ in $\mathcal{B}$ is finite-dimensional modulo its Jacobson radical. Finally, we point out that one of our main results can be applied to the implementation of local unitary (LU) equivalence of quantum states.

math.FA↗
arXiv2026

Bargmann invariants and local unitary equivalence

In this paper, we study the local unitary equivalence of quantum states on $\mathbb{C}^n\otimes\mathbb{C}^n$, which is an important notion in quantum information theory. For the case $n=2$, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for $n\geqslant 3$, which negatively answers a problem proposed by L. Zhang and B. Xie.

quant-ph↗
arXiv2026

Think Thrice Before You Speak: Dual knowledge-enhanced Theory-of-Mind Reasoning for Persuasive Agents

Persuasive dialogue requires reasoning about others' latent mental states, a capability known as Theory of Mind (ToM). However, due to reliance on simple prompting strategies and insufficient ToM knowledge, existing LLMs often fail to capture the intrinsic dependencies among mental states, leading to fragmented representations and unstable reasoning. To address these challenges, we introduce the ToM-based Persuasive Dialogue (ToM-PD) task, grounded in the Belief-Desire-Intention (BDI) framework, which explicitly models the sequential dependencies among mental states in multi-turn dialogues. To facilitate research on this task, we construct a large-scale annotated dataset, ToM-based Broad Persuasive Dialogues (ToM-BPD), capturing fine-grained mental states and corresponding persuasive strategies. We further propose Think Thrice Before You Speak (TTBYS), a knowledge-enhanced stepwise reasoning framework that leverages both explicit and implicit prior experiences to improve LLMs' inference of desires, beliefs, and persuasive strategies. Experimental results demonstrate that Qwen3-8B equipped with TTBYS outperforms GPT-5 by 1.20%, 22.80%, and 16.97% in predicting desires, beliefs, and persuasive strategies, respectively. Case studies further show that our approach enhances interpretability and consistency in reasoning.

cs.AI↗
arXiv2026

The similarity of irreducible operators in factors

An operator $T$ in a separable factor $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann subalgebra $W^*(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(T)'\cap\mathcal{M}=\mathbb{C}I$. We say that $T$ is a single generator of $\mathcal{M}$ if $W^*(T)=\mathcal{M}$. In this paper, we study generators of separable factors related to maximal abelian self-adjoint subalgebras. As an application, we obtain a complete characterization of normal operators in separable factors which are similar to irreducible operators.

math.OA↗
arXiv2026

Density of irreducible operators in the trace-class norm

In 1968, Paul Halmos initiated the research on density of the set of irreducible operators on a separable Hilbert space. Through the research, a long-standing unsolved problem inquires: is the set of irreducible operators dense in $B(H)$ with respect to the trace-class norm topology? Precisely, for each operator $T $ in $B(H)$ and every $\varepsilon >0$, is there a trace-class operator $K$ such that $T+K$ is irreducible and $\Vert K \Vert_1 < \varepsilon$? For $p>1$, to prove the $\Vert \cdot \Vert_p$-norm density of irreducible operators in $B(H)$, a type of Weyl-von Neumann theorem effects as a key technique. But the traditional method fails for the case $p=1$, where by $\Vert \cdot \Vert_p$-norm we denote the Schatten $p$-norm. In the current paper, for a large family of operators in $B(H)$, we give the above long-term problem an affirmative answer. The result is derived from a combination of techniques in both operator theory and operator algebras. Moreover, we discover that there is a strong connection between the problem and another related operator-theoretical problem related to type $\mathrm{II}_1$ von Neumann algebras.

math.OA↗
arXiv2026

Operators with disconnected spectrum in von Neumann algebras

Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and $Φ$ a unitarily invariant $\|\cdot\|$-dominating norm on $\mathcal{I}$. In this paper, we provide a necessary and sufficient condition on $Φ$ such that every operator in $\mathcal{M}$ can be expressed as the sum of an operator in $\mathcal{M}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose $Φ$-norm is arbitrarily small. Similarly, if $\mathcal{A}$ is a unital $C^*$-algebra of real rank zero with dimension greater than one and $\mathcal{I}$ is an essential ideal in $\mathcal{A}$, then every element in $\mathcal{A}$ can be written as the sum of an operator in $\mathcal{A}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose norm is arbitrarily small.

math.OA↗
arXiv2025

Products of irreducible operators in factors

Let $\mathcal M$ be a separable factor. An operator $T$ in $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann algebra $W^*(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(T)'\cap\mathcal{M}=\mathbb{C}I$. In this paper, we show that every operator in a separable factor $\mathcal{M}$ is the product of two irreducible operators in $\mathcal{M}$, except the zero operator in factors of type $\mathrm{I}_{2n+1}$ for $n\geqslant 1$. This may be viewed as a multiplicative analogue of Radjavi's result which asserts that every operator on a separable Hilbert space is the sum of two irreducible operators.

math.OA↗
arXiv2025

Ortho-isomorphisms of von Neumann algebras

Suppose $\mathscr M$ and $\mathscr N$ are von Neumann algebras. Two operators $A$ and $B$ in $\mathscr M$ are said to be orthogonal if $A^*B=0$, meaning their ranges are orthogonal. Let $φ\colon\mathscr M\to\mathscr N$ be a map. We say that $φ$ is an ortho-isomorphism if it is bijective and satisfies that $A^*B=0$ if and only if $φ(A)^*φ(B)=0$ for all $A,B\in\mathscr M$. The map $φ$ is called ortho-additive if the additive relation $φ(A+B)=φ(A)+φ(B)$ holds for all $A,B\in \mathscr M$ with $A^*B=0$. In this paper, we characterize the complete structure of ortho-additive ortho-isomorphisms between von Neumann algebras, which is an analogue of Dye's theorem and Uhlhorn's theorem.

math.OA↗
arXiv2025

Irreducible operators in von Neumann algebras

Let $\mathcal{M}$ be a separable von Neumann algebra with center $\mathcal{Z}(\mathcal{M})$. An operator $T$ in $\mathcal{M}$ is called irreducible if the von Neumann algebra $W^*(T)$ generated by $T$ has trivial relative commutant, i.e., $W^*(T)'\cap\mathcal{M}=\mathcal{Z}(\mathcal{M})$. In this paper, we show that irreducible operators in $\mathcal{M}$ form a norm-dense $G_δ$ set, which is a generalization of Halmos' theorem. Moreover, we prove that every operator in $\mathcal{M}$ is the sum of two irreducible operators, which is an analogue of Radjavi's theorem.

math.OA↗
arXiv2025

Voiculescu's Theorem in Properly Infinite Factors

In this paper, we investigate Voiculescu's theorem on approximate unitary equivalence in separable properly infinite factors. As applications, we establish the norm-denseness of the set of all reducible operators, prove a generalized Voiculescu's bicommutant theorem and a version of asymptotic bicommutant theorem, and obtain an interesting cohomological result. Additionally, we extend these results to multiplier algebras within separable type $\mathrm{III}$ factors. At last, a concept of the nuclear length is introduced.

math.OA↗
arXiv2025

Free semigroupoid algebras and the first cohomology groups

This paper investigates derivations of the free semigroupoid algebra $\mathfrak{L}_G$ of a countable or uncountable directed graph $G$ and its norm-closed version, the tensor algebra $\mathcal{A}_G$. We first prove a weak Dixmier approximation theorem for $\mathfrak{L}_G$ when $G$ is strongly connected. Using the theorem, we show that if every connected component of $G$ is strongly connected, then every bounded derivation $δ$ from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is of the form $δ=δ_T$ for some $T\in\mathfrak{L}_G$ with $\|T\|\leqslant\|δ\|$. For any finite directed graph $G$, we also show that the first cohomology group $H^1(\mathcal{A}_G,\mathfrak{L}_G)$ vanishes if and only if every connected component of $G$ is either strongly connected or a fruit tree. To handle infinite directed graphs, we introduce the alternating number and propose \Cref{conj intro-in-tree}. Suppose every connected component of $G$ is not strongly connected. We show that if every bounded derivation from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is inner, then every connected component of $G$ is a generalized fruit tree and the alternating number $A(G)$ of $G$ is finite. The converse is also true if the conjecture holds. Finally, we provide some examples of free semigroupoid algebras together with their nontrivial first cohomology groups.

math.OA↗
arXiv2025

On cleanness of AW*-algebras

A ring is called clean if every element is the sum of an invertible element and an idempotent. This paper investigates the cleanness of AW*-algebras. We prove that all finite AW*-algebras are clean, affirmatively solving a question posed by Vas. We also prove that all countably decomposable infinite AW*-factors are clean. A *-ring is called almost *-clean if every element can be expressed as the sum of a non-zero-divisor and a projection. We show that an AW*-algebra is almost *-clean if and only if it is finite.

math.OA↗