TITLE: If two projections are close, then they are unitarily equivalent
QUESTION [13 upvotes]: Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$. 
The proof that immediately occurs to me uses comparison of projections (the inequality forces $p\sim q$ and also $p^\perp\sim q^\perp$, and then one constructs the unitary by adding the corresponding partial isometries), and implies that one can even replace $B(H)$ with any von Neumann algebra that contains both $p$ and $q$. 

Questions:

Does anyone know of a more direct proof? Comparison of projections is a very basic result in von Neumann algebras, but somehow here it feels like overkill. 
Does this property hold in a C$^*$-algebra? Proof, or counterexample?

REPLY [4 votes]: In Two Subspaces (Trans. AMS 1969), Halmos made some useful observations about pairs of subspaces/projections. When $\|p-q\|<1$, this yields in particular that we can assume, up to unitary equivalence, that
$$
p=\pmatrix{1&0\\0&0} \qquad q=\pmatrix{c^2& cs\\cs& s^2}\qquad c^2+s^2=1
$$
where $c, s$ are commuting positive contractions which allow computations just like for pairs of rank one projections in the two-dimensional case where $c=\cos \theta$ and $s=\sin\theta$. Then it is natural to come up with the following candidate for the unitary you want, namely
$$
u=\pmatrix{c & s\\s&-c}
$$
Once we have found that, we just have to express this in terms of $p$ and $q$. And we find

$$
u=(p+q-1)|p+q-1|^{-1}
$$ 
  which is a self-adjoint involution, lying obviously in the $C^*$-algebra geberated by $p$ and $q$. 

Note  To check that the above formula does the job, the key point is that $(p+q-1)^2=1-(p-q)^2$ is invertible, positive, and commutes with $p$ and $q$.
Remark   Another candidate would be the "rotation" 
$$
v=\pmatrix{c& -s\\ s& c}\qquad \mbox{i.e.} \quad v=u(2p-1)
$$
An advantage of the latter unitary is that we have 
$$
\|v-1\|\leq \sqrt{2}\|p-q\|
$$
If I recall correctly, this is mentioned in Takesaki I for reference purposes.