Bochner integral of functions valued in a C*algebra

Question

Let $A$ be a C*-algebra and let $f\in L^1(\mathbb{R},A)$ be a Bochner integrable function. Suppose that $f(t)\geq 0$ for all $t\in\mathbb{R}$ and that $\int f=0$. Does that imply that $f$ vanishes almost everywhere? I can see that for every $\lambda$ in the Banach space dual of $A$, $\lambda \circ f$ vanishes everywhere, but I don't think that suffices.

Answer 1

Since $f$ is BI its is almost separably valued: There is a separable subspace $M$ of $A$ and a null set $E$ such that $f(x) \in M$ if $x \notin E$.

The closed unit ball of $M^{*}$ is a compact metric space and hence also separable in the weak* topology. If $(\lambda_i)$ is a countable dense subset in this ball then $\lambda_i\circ f=0$ almost eveywhere for each $i$. Hence there is one null set outside which $\lambda_i\circ f=0$ for each $i$. This implies $\lambda\circ f=0$ for each $\lambda$ in the close unit ball, hence also for each $\lambda \in M^{*}$ outside this null set. It follows that $f=0$ a.e.