Continuity of traces on uniform tracial completions
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Must every trace on a uniform tracial completion be continuous in the uniform 2-norm? A related formulation asks the same for factorial tracially complete pairs.
Why it matters and what counts as progress
Why it matters: Continuity controls whether completion preserves the intended tracial structure without introducing discontinuous traces.
What counts as progress: A continuity theorem or an explicit completion or factorial pair carrying a discontinuous trace, with the relevant uniform norm and compact trace-space hypotheses retained.
How the problem developed
- 2025 · Open problem [1]
Schafhauser, Tikuisis and White Posed Problem XXII: whether every trace on the uniform tracial completion is continuous in the uniform 2-norm.
- 2025 · Partial result [2]
Samuel Evington Solved the trace problem positively for C*-algebras that absorb the Jiang–Su algebra.
Scott Sauers and Astra A registry-recorded construction gives a separable unital example with a discontinuous trace, and a factorial tracially complete pair with the same failure.
AI contribution
Astra-origin construction gives discontinuous traces
Sauers constructs a separable unital C*-algebra with compact nonempty trace space whose uniform tracial completion has a trace discontinuous in the uniform 2-norm, and a factorial tracially complete pair with the same failure.
Sources
- Schafhauser, Tikuisis and White, Nuclear C*-algebras: 99 problems
- Evington, Traces on the uniform tracial completion of Z-stable C*-algebras
- Palomar, A Counterexample to the Trace Problem for Uniform Tracial Completions
- Sauers, group-approximation research repository
- Sauers, discontinuous-trace construction and verification
- Schafhauser, Tikuisis and White, Problem XXII
- Carrion and coauthors, factorial-pair question
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