= Solution
Write $p:U(H)\to PU(H)$ for the scalar quotient. Local strongly continuous sections exist: near a fixed projective unitary, fix its phase by requiring one nonzero <matrix> coefficient, taken relative to a fixed representative, to be positive real. Thus the pullback $E=\{(z,V):p(V)=\rho(z)\}$ is a locally trivial central extension of the <circle> by the <circle>. It is compact, since finitely many closed arcs inside such trivializing neighbourhoods cover the compact base and each pullback is a product with the compact fibre.
Any two lifts commute up to a scalar. Their <commutator> defines a continuous alternating bicharacter $c(z,w)$ on the base: the scalar <commutator> is independent of phase choices and multiplicative in each argument. If $z$ is an irrational rotation, its powers are dense and $c(z,z^m)=1$ for every integer $m$. Continuity makes $c(z,w)=1$ for every $w$. Such $z$ are themselves dense, so \b[all the lifts commute], and $E$ is abelian.
The tautological representation $R(z,V)=V$ is strongly continuous. Choose a unit <vector> $\xi$ and average its <rank-one orthogonal projection> over <Haar measure> on $E$:
$$
K=\int_E R(e)|\xi\rangle\langle\xi|R(e)^*\,de.
$$
The integrand is trace-norm continuous, so $K$ is positive compact with <trace> one, and commutes with $R(E)$. A positive <eigenvalue> has a nonzero finite-dimensional <eigenspace> invariant under $E$. Commuting unitary <matrices> on that space have a common unit <eigenvector> $\eta$. Therefore $R(e)\eta=\chi(e)\eta$ for a continuous character $\chi$ of $E$.
On the central fibre, $R(1,\lambda I)=\lambda I$, so $\chi(1,\lambda I)=\lambda$. Consequently
$$
\boxed{U_z=\chi(z,V)^{-1}V\quad\text{for any lift }V\text{ of }\rho(z)}
$$
is independent of that lift. It is a <homomorphism> and strongly continuous, because the continuous map on $E$ descends through its quotient to the base. It satisfies $p(U_z)=\rho(z)$. This proves <circle projective representations lift to unitary representations> using an actual central character, rather than assuming a global choice of phases.
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