Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-algebras

Samuel Evington

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arXiv · 2407.00727

Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-algebras

Samuel Evington

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Abstract

The uniform tracial completion of a C*-algebra A with compact non-empty trace space T(A) is obtained by completing the unit ball with respect to the uniform 2-seminorm $\|a\|_{2,T(A)}=\sup_{\tau \in T(A)} \tau(a^*a)^{1/2}$. The trace problem asks whether every trace on the uniform tracial completion is the $\|\cdot\|_{2,T(A)}$-continuous extension of a trace on A. We answer this question positively in the case of C*-algebras that tensorially absorb the Jiang-Su algebra, such as those studied in the Elliott classification programme.

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Samuel Evington. 2024-06-30. Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-algebras. https://arxiv.org/abs/2407.00727

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