 Solution

First, all lifts of the real one-parameter projective group commute. Its scalar commutator $c(s,t)$ is again a continuous alternating bicharacter. For fixed nonzero $t$ and an integer $n>0$, a lift of $t$ is a phase multiple of the $n$th power of a lift of $t/n$; hence $c(t,t/n)=1$ and $c(t,mt/n)=1$ for every integer $m$. Rational multiples of $t$ are dense in $\mathbb{R}$, so continuity makes $c(t,s)=1$ for every $s$.

Choose a unitary $U$ representing $\rho (1)$ and the allowed self-adjoint logarithm $A\in U^{\prime \prime }$ with $U=e^{iA}$. Every lift of $\rho (t)$ commutes with $U$, hence with $A$ by the double commutant theorem for star-algebras condition. Thereforeis a continuous projective homomorphism, with $\tilde{\rho }(1)=1$. It is periodic of period one and descends to $\mathbb{R}/\mathbb{Z}$. Part (b) lifts it to a continuous unitary homomorphism $L$ of that circle. Each $L(t)$ is a lift of $\tilde{\rho }(t)$ and commutes with $A$, since a representative has the form of an original lift times $e^{-itA}$. Thusis a strongly continuous unitary homomorphism with projective class $\rho (t)$. This gives the required lift of the whole real group, not just unrelated unitary representatives at each time.