Complex structures on hyperplanes of the Kalton–Peck space
Follow the latest progress by email.
Is the real Kalton–Peck space Z₂ isomorphic to any of its hyperplanes? A stronger obstruction asks whether any hyperplane admits a complex structure.
Why it matters and what counts as progress
Why it matters: It tests a basic infinite-dimensional Banach-space isomorphism property in a distinguished twisted-sum example.
What counts as progress: Prove that hyperplanes admit no complex structure, or construct an isomorphism with a hyperplane; retain the real-space setting.
How the problem developed
- 1979 · Space construction [1]
Nigel J. Kalton and N. T. Peck Introduced the twisted Hilbert space Z₂ that became the subject of the hyperplane isomorphism question.
- 2014 · Conjecture [2]
Valentin Ferenczi and collaborators Recorded the conjecture that Z₂ is an even Banach space: it has a complex structure, but none of its hyperplanes does.
- 3 Aug 2026 · Resolution [3]
A. Das, Valentin Ferenczi and Christian Rosendal Proved that no hyperplane of the real Z₂ admits a complex structure, resolving the conjecture that Z₂ is not isomorphic to its hyperplanes.
AI contribution
Kalton-Peck space is not isomorphic to its hyperplanes
Das, Ferenczi and Rosendal prove that no hyperplane of the real Kalton–Peck space Z₂ admits a complex structure; hence Z₂ is not isomorphic to its hyperplanes. A symplectic operator rank-parity theorem supplies the obstruction.
Sources
- Kalton and Peck · Twisted sums of sequence spaces and the three-space problem (1979)
- González and Pello · Open problems on hyperplanes and complex structures (2014)
- Das, Ferenczi and Rosendal · Kalton–Peck space is not isomorphic to its hyperplanes (2026)
- Das, Ferenczi and Rosendal, Kalton–Peck hyperplanes
- Kalton–Peck Lean formalization
Website timeline
Publication dates and editorial changes.