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Say $A$ is an exact C*-algebra and let $T(A)$ be the cone of densely defined lower semicontinous traces. It is known that if $a \in \mathrm{Ped}(A)$ is full, then $T_{a\to 1} := \{\tau \in T(A) \mid \tau(a) = 1\}$ is a base of $T(A)$.

Is the converse true?

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Let $A := \mathcal O_2 \oplus \mathbb C$. Then, there is a unique tracial state given by $$ \tau(a,\lambda) := \lambda. $$ Let $a := (0,1) \in A$. Then clearly $T_{a \mapsto 1}$ is a base for $T(A)$ but $a$ is not full.

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  • $\begingroup$ I guess what I really wanted to ask was for stably finite algebras. $\endgroup$ – Nick Bottom Sep 28 '18 at 16:04

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