# 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.

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