The 4th Barcelona Weekend on Operator Algebras: Open Problems, New Results, and Community

The 4th Barcelona Weekend on Operator Algebras, held at the CRM on January 30–31, 2026, brought together experts to discuss recent advances and open problems in the field.
The event strengthened the exchange of ideas within the community and reinforced the CRM’s role as a key meeting point for research in C*-algebras and related areas.

The Centre de Recerca Matemàtica (CRM) hosted the 4th Barcelona Weekend on Operator Algebras on January 30–31, 2026. The meeting was devoted to the algebraic theory of operator algebras, with a particular focus on C*-algebras, their invariants, and their interactions with group theory, dynamics, and mathematical logic.

The aim of the event was to bring together international experts and early-career researchers to share recent advances, foster new connections between research lines, and further consolidate this workshop as a reference meeting point for the operator algebras community.

Talks and speakers

• Fernando Lledó (Universidad Carlos III de Madrid) opened the program with KMS States for Separated Graph Algebras – I, introducing a measure-theoretic approach to equilibrium states and laying the conceptual foundations for the topic.
• Shanshan Hua (University of Münster) examined Cartan Subalgebras of Self-Similar Graph Algebras, extending known descriptions beyond the locally faithful setting and constructing new groupoid models that recover part of the original self-similar dynamics.
• Tatiana Shulman (University of Gothenburg) spoke on Cones and Asymptotic Liftings, focusing on cone C*-algebras and analysing both established and new properties—such as quasidiagonality and homotopy invariance—and their role in understanding deep algebraic phenomena.
• Conchita Martínez (Universidad de Zaragoza) addressed Coherence and Derived Groups of Artin Groups, showing that the characterization of coherence in terms of the derived group holds in full generality and discussing its still poorly understood relationship with the coherence of group rings.
• Eduard Vilalta (Universitat Politècnica de Catalunya) presented Strict Comparison for Twisted Group C*-Algebras, highlighting recent progress on strict comparison and its versatility, with applications ranging from classification results to classical problems in group theory.
• Guillem Quingles (Universitat Autònoma de Barcelona) focused on The Cuntz Semigroup of Rings of Continuous Functions on One-Dimensional Spaces, describing this invariant in terms of monoids of intervals and revealing notable differences between real-valued and complex-valued function rings.
• Apurva Seth (University of Oxford) concluded the first day with The Radius of Comparison for Actions on Simple AH Algebras, constructing explicit actions of on simple C*-algebras that allow precise control of the radius of comparison before and after forming crossed products.
• Joachim Zacharias (University of Glasgow) reported on AF-Embeddability for Decomposition Rank One Algebras, addressing key structural questions relevant to the classification program for C*-algebras.
• Ben Bouwen (Czech Academy of Sciences) discussed Embeddings into the Ultrapower of the Jiang–Su Algebra, a central problem in the model theory of C*-algebras related to existential closure, showing that cones over separable C*-algebras embed into the ultrapower of the Jiang–Su algebra and proposing this embeddability as a new regularity property.
• Joan Claramunt (Universitat Politècnica de Catalunya) closed the event with KMS States for Separated Graph Algebras – II, presenting a second method to construct KMS states for general separated graph C*-algebras and broadening the range of available techniques for studying these systems.

A space for mathematical exchange

The CRM is particularly pleased to have hosted this fourth edition of the Barcelona Weekend on Operator Algebras, which has strengthened its role as a high-level forum for scientific exchange, encouraged dialogue across different algebraic perspectives, and reinforced the ties of an active, diverse, and steadily growing international research community.