算子代数：2025–2026 年度综述

An annual survey of the math.OA arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• C*-Algebra Structure and K-Theory (14.5%) — Investigates structural properties, isomorphism problems, and K-theoretic invariants of general C*-algebras, including graph algebras and ultraproducts.
• Von Neumann Algebra Subfactor Theory (13.2%) — Focuses on the classification, entropy, and rigidity properties of von Neumann algebras, factors, and their subalgebras.
• Group Actions and Crossed Products (11.9%) — Analyzes C*-algebras arising from group actions, crossed products, and amenability properties of group representations.
• Free Probability and Random Operators (10.8%) — Explores noncommutative probability, free independence, and spectral distributions of random operators and matrix models.
• Quantum Groups and Quantum Graphs (10.1%) — Studies compact quantum groups, quantum graph algebras, and their applications in quantum information and transport.
• Groupoid C*-Algebras and Homology (9.9%) — Examines étale and ample groupoids, their associated C*-algebras, and homological invariants in topological dynamics.
• Operator Systems and Completely Positive Maps (7.9%) — Investigates the geometry and structure of operator systems, completely positive maps, and their associated metrics.
• Spectral Theory and Toeplitz Operators (7.8%) — Addresses spectral properties of operators, including Toeplitz operators, spectral triples, and geometric deformations.
• Banach Algebras and Cocycle Dynamics (7.3%) — Covers twisted crossed products of Banach algebras, intertwining cocycles, and noncommutative differentiability.
• Spectral Triples and Quantum Group Representations (6.7%) — Focuses on spectral triples, isotypic decompositions, and amenability of locally compact quantum groups.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.