Mathematicians are building a complete family tree for a class of abstract algebraic structures called C*-algebras, aiming to sort them into tidy, predictable categories. These structures emerge naturally wherever quantum mechanics, group symmetries, or dynamical systems meet Hilbert spaces—the infinite-dimensional vector spaces that underpin quantum theory. For decades, researchers have had a near-perfect classification for a related family, von Neumann algebras, thanks to Alain Connes’s 1970s breakthroughs. But the C*-algebra version—the topological cousin—has remained stubbornly incomplete. This project aims to close that gap. If successful, the work will produce definitive classification theorems for amenable C*-algebras and their morphisms, and will launch a systematic study of quantum symmetries within this framework. This is fundamental mathematics: there is no immediate practical application. But the von Neumann classification it seeks to mirror has powered decades of progress in measurable dynamics, subfactor theory, and rigidity problems—areas that quietly shape how physicists model quantum systems and how mathematicians understand symmetry. A C*-algebra analogue could unlock similarly deep, unexpected insights.
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Operator algebras arise as families of bounded operators on a Hilbert space, closed under algebraic operations, the Hilbert space adjoint, and under taking limits. There are two main types of operator algebra - von Neumann algebras and C*-algebras - which are closed under pointwise and uniform convergence respectively. Both through the structure of abelian algebras, and also the tools used to work with them, von Neumann algebras have the flavour of measure theory, while C*-algebras are topological in nature. Examples arise wherever mathematics touches Hilbert spaces, so operator algebras occur naturally from group representations, dynamics, mathematical physics, to name but a few. The central theme of this proposal is the structure and classification of amenable C*-algebras, with an emphasis on examples coming from dynamics. This aims for the full C*-algebra analogue of Connes' groundbreaking advances in the structure theory of von Neumann algebras from the 1970's which led to the complete classification of amenable von Neumann algebras (the Connes-Haagerup classification of injective factors) and has remained critical ever since, powering dramatic subsequent developments in measurable dynamics, subfactors and rigidity problems. The proposal seeks to obtain definitive classification theorems for amenable morphisms between C*-algebras together with powerful structure theorems which identify classifiable algebras and morphisms both abstractly and in important families of examples. This will be used to initiate a deep study of quantum symmetries of amenable C*-algebras through a classification of actions of tensor categories, aiming for the profound impact seen in the von Neumann algebraic framework through Jones theory of subfactors. A driving theme throughout the project is the explicit transfer of ideas and techniques from the von Neumann algebraic framework to C*-algebras: the use of von Neumann techniques in C*-algebras.
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