Which matrices represent unitary projective representations of ${\rm SO(3)}$?

Question

I was reading this post which triggered the following question.

The group ${\rm SO(3)}$ is real orthogonal. However, it is possible to consider representations of ${\rm SO(3)}$ on a complex vector space. In particular, any element $R_a\in {\rm SO(3)}$ can be represented by a unitary matrix $U(R_a)$ by choosing the generators $J_i$ to be hermitian and satisfying the Lie algebra $$[J_a,J_b]=i\epsilon_{abc}J_b.\tag{1}$$ For example, choosing $J_a=\frac{1}{2}\sigma_a$, we obtain a Unitary representation $$R_a\mapsto U(R_a)=\exp\Big[\frac{-i\theta_a \sigma_a}{2}\Big]\tag{2}.$$ This is a two dimensional Unitary representation with $j=1/2$. We can also arrive at higher dimensional unitary representations of with $j=3/2,5/2,...$ etc.

$1.$ What should I call the representation $(2)$ (and all those with $j=3/2,5/2,...$) ? Should I call it a unitary representation of ${\rm SO(3)}$ or call it the fundamental unitary representation of ${\rm SU(2)}$?

$2.$ If $(2)$ is a complex unitary representation of ${\rm SO(3)}$, is it projective? I do not see that because $$R_z(\pi)R_z(\pi)=R_z(2\pi)\tag{3}$$ is mapped to $$U(R_z(\pi))U(R_z(\pi))=U(R_z(2\pi))$$ which looks non-projective or ordinary. However, according to Wikipedia, half-integral spin representations are projective representations of ${\rm SO(3)}$.

$3.$ If $(2)$ and its higher-dimensional representations with $j=3/2,5/2,...$ etc are all ordinary non-projective representations of ${\rm SO(3)}$, it seems that it is not necessary to talk about the ${\rm SU(2)}$ group in quantum mechanics. Apparently, we shall not lose any physics by confining ourselves to the ordinary non-projective complex unitary representations of ${\rm SO(3)}$.

Answer 1

1. In the defining 3-dimensional representation of the 3D rotation group $SO(3)$ a $2\pi$-rotation is the identity element $$R(2\pi)~=~\mathbb{1}_{3\times 3}.\tag{1}$$

2. In fact, the identity element of a group is represented by the unit matrix for any group representation.

3. For $2\times 2$-matrices, we calculate using various properties of the Pauli matrices that $$U(R(2\pi))~=~-\mathbb{1}_{2\times 2},\tag{2}$$ which shows that $U$ cannot be a genuine group representation of $SO(3)$.