A question about finite-rank projection

Question

Let $B, C$ be two C*-algebras and $\sigma_{0}: B\rightarrow C$ be *-homomorphism such that $\sigma_{0}$ is injective. Then, for a finite set $F\subset B$ of the unit ball and $\varepsilon>0$,

Can we find a finite-rank projection $Q$ such that $$||Q\sigma_{0}(b)Q||\geq||b||-\varepsilon$$ for all $b\in F$?

Answer 1

Certainly not in that generality. For instance, $C$ could have no finite-rank projections at all; it could be even projectionless.

Even in cases where $C$ has finite-rank projections, what you want can fail, as the finite-rank projections may exists only in a direct summand.