Kirchberg's embedding problem
Kirchberg's embedding problem: does every separable C*-algebra embed into the norm ultrapower of the Cuntz algebra O₂? The answer is no: the full C*-algebra of an explicit group has no unital embedding into the norm ultrapower of any nuclear C*-algebra. The proof: a spectral test for representations of a group with property (T), a finiteness statement in nuclear algebras, and an endomorphism of finite index that forces chains of representations. Proved in full from two named hypotheses: property (T) of the group ℤ³ ⋊ SL₃(ℤ), and the nuclearity of O₂. The course is an exposition of OpenAI's proof (2026).
Three lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Norm ultrapowers and a group with property (T)
- A spectral test and nuclear finiteness
- Predecessors and the contradiction
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Tensor positivity and nuclearity (Positive maps and finite-dimensional approximation)
- Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
- Compact and trace-class operators, the predual of B(H), and the operator topologies · Foundations of von Neumann algebras
- Hilbert spaces and compact operators · Foundations of von Neumann algebras
- Building representations from positive functionals · Foundations of von Neumann algebras
- C*-algebras and trace-class operators (course Foundations of von Neumann algebras); the minimal and maximal tensor products (course Positive maps and finite-dimensional approximation).