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Motivated by classical investigation of conjugation invariant positive-definite functions on discrete groups, we study tracial central states on universal C*-algebras associated with compact quantum groups, where centrality is understood in the sense of invariance under the adjoint action. We fully classify such states on $q$-deformations of compact Lie groups, on free orthogonal quantum groups, quantum permutation groups and on quantum hyperoctahedral groups., Comment: 29 pages, v2 corrects a few minor points

The $q$-Araki-Woods factor associated to a group of orthogonal transformations on a real separable Hilbert space $\mathsf{H}_{\mathbb{R}}$ is full as soon as $\dim\mathsf{H}_{\mathbb{R}}\geq 2$., Comment: 18 pages; minor corrections; final version to appear in JLMS

Recent advancements in recommendation systems have shifted towards more comprehensive and personalized recommendations by utilizing large language models (LLM). However, effectively integrating LLM's commonsense knowledge and reasoning abilities into recommendation systems remains a challenging problem. In this paper, we propose RecSysLLM, a novel pre-trained recommendation model based on LLMs. RecSysLLM retains LLM reasoning and knowledge while integrating recommendation domain knowledge through unique designs of data, training, and inference. This allows RecSysLLM to leverage LLMs' capabilities for recommendation tasks in an efficient, unified framework. We demonstrate the effectiveness of RecSysLLM on benchmarks and real-world scenarios. RecSysLLM provides a promising approach to developing unified recommendation systems by fully exploiting the power of pre-trained language models., Comment: 13 pages, 4 figures

Recommendation systems aim to provide users with relevant suggestions, but often lack interpretability and fail to capture higher-level semantic relationships between user behaviors and profiles. In this paper, we propose a novel approach that leverages large language models (LLMs) to construct personalized reasoning graphs. These graphs link a user's profile and behavioral sequences through causal and logical inferences, representing the user's interests in an interpretable way. Our approach, LLM reasoning graphs (LLMRG), has four components: chained graph reasoning, divergent extension, self-verification and scoring, and knowledge base self-improvement. The resulting reasoning graph is encoded using graph neural networks, which serves as additional input to improve conventional recommender systems, without requiring extra user or item information. Our approach demonstrates how LLMs can enable more logical and interpretable recommender systems through personalized reasoning graphs. LLMRG allows recommendations to benefit from both engineered recommendation systems and LLM-derived reasoning graphs. We demonstrate the effectiveness of LLMRG on benchmarks and real-world scenarios in enhancing base recommendation models., Comment: 12 pages, 6 figures

We describe explicitly all actions of the quantum permutation groups on classical compact spaces. In particular, we show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated to non-crossing partitions., Comment: 21 pages. v2, v3: minor changes

We prove a pointwise ergodic theorem for actions of amenable groups on noncommutative measure spaces. To do so, we establish the maximal ergodic inequality for averages of operator-valued functions on amenable groups. The main arguments are a geometric construction of martingales based on the Ornstein-Weiss quasi-tilings and harmonic analytic estimates coming from noncommutative Calder\'on-Zygmund theory., Comment: Preliminary version, comments are welcome. 22 pages

The $q$-deformed Araki-Woods von Neumann algebras $\Gamma_q(\mathcal{H}_\mathbb{R}, U_t)^{\prime \prime}$ are factors for all $q\in (-1,1)$ whenever $dim(\mathcal{H}_\mathbb{R})\geq 3$. When $dim(\mathcal{H}_\mathbb{R})=2$ they are factors as well for all $q$ so long as the parameter defining $(U_t)$ is `small' or $1$ $($trivial$)$ as the case may be., Comment: 25 pages

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang., Comment: v3: some corrections in Section 3,4 and 6; final version to appear in CMP