Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak star closed subalgebra of $M$), $T(\phi(x))=T(x) \quad \forall x \in M$.

Does this imply that ,
$T(\phi(x))=T(x) \quad \forall x \in L^{1}(M)$ ?
where $L^{1}(M)$ is the completion of $M$ in $||.||_{1}$.

I know that we can extend $\phi$ continuously to $L^{1}(M)$. How can I use this fact to answer my question?


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