Characters of central sequences and Jiang–Su stability
Kirchberg and Rørdam asked whether a unital separable C*-algebra A absorbs the Jiang–Su algebra Z whenever its central sequence algebra F_ω(A) has no characters. The answer is yes. The core is a finite theorem: every nonzero unital C*-algebra without characters has a finite maximal tensor power that receives a unital homomorphism from the dimension-drop algebra I(2,3). Its proof: the Rørdam–Winter criterion in an arbitrary unital C*-algebra; labels that stay linearly independent in every irreducible representation of a tensor power; a full square-zero element and a homomorphism from the matrix cone; a cube model of tensor powers of the cone. Proved in full from two named hypotheses: the theorem of Kirchberg and Rørdam relating Z-stability to F_ω(A), and (for infinite tensor powers) the theorem of Dadarlat and Toms. The course is an exposition of OpenAI's proof (2026).
Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Central sequences and the character criterion
- Cuntz comparison and order-zero maps
- The Rørdam–Winter criterion
- Labels that survive representations
- A full cone in a tensor power
- From the cone to the dimension-drop algebra
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Regular values and degree modulo two (Borsuk's conjecture fails in dimension nine)
- Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
- Building representations from positive functionals · Foundations of von Neumann algebras
- The double commutant theorem · Foundations of von Neumann algebras
- The universal enveloping von Neumann algebra of a \(C^*\)-algebra, and \(W^*\)-algebras · Foundations of von Neumann algebras
- Norm ultrapowers and a group with property (T) (Kirchberg's embedding problem)
- Tensor products of Banach and Hilbert spaces, Jordan homomorphisms and isometries of \(C^*\)-algebras · Tensor products of operator algebras
- Tensor products of C\*-algebras and the minimal norm · Tensor products of operator algebras
- C*-algebras, representations, the double commutant theorem and the universal enveloping von Neumann algebra (course Foundations of von Neumann algebras); tensor products of C*-algebras (course Tensor products of operator algebras); Sard's theorem (lesson Regular values and degree modulo two).