A Counterexample to the Trace Problem for Uniform Tracial Completions
We formalize a negative answer to Problem XXII of Schafhauser, Tikuisis and White. There is a separable unital C*-algebra A with nonempty compact tracial state space such that its uniform tracial completion M has a tracial state that is discontinuous in the uniform 2-norm. Thus the canonical extensions of the traces of A form a proper subset X of the traces of M. A second theorem gives a factorial tracially complete pair (M, X) with the same failure, answering Question 1.1 of Carrion, Castillejos, Evington, Gabe, Schafhauser, Tikuisis and White negatively. In both statements, X is exactly…
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