Projections on the GNS Hilbert space of a pure state

Question

Given a $C*$ algebra A and a pure state $f$ with the GNS construction $(\Pi, H, \Omega_f)$ such that $\Pi(A)''=B(H)$.

1. Does a finite projection to any 1d subspace of $H$ lie in $\Pi(A)$?

2. What are the elements in $A$ that gets mapped to projections in $B(H$) by the GNS representation $\Pi$.

Answer 1

Take $A$ to be any properly infinite C$^*$-algebra, or a projectionless one. Represent it irreducibly. Any irrep comes from a pure state, so you get an example where $\pi(A)$ has no finite projections, or even no projections at all.

The above also answers 2: if you take $A$ to be projectionless, say $A=C_r^*(\mathbb F_2)$, then $\pi(A)$ contains no projections.