Bases of the tracial cone and full elements 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?

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.