Partial isometry and perturbation of a $C^*$ algebra. Let $A$ be a $C^*$algebra and $I$ be an ideal. Suppose that $I$ admits an approximate identity consisting of projections in $I$. Let $u \in U(A/I)$. Then is there a partial isometry $v \in A$ such that $\pi(v)= u$, where $\pi$ is the quotient map?
I think if we use perturbation we can find such partial isometry but I couldn't. Could anybody help me out?
 A: Given $u$ in $U(A/I)$, choose any $w$ in $A$ such that $\pi (w)=u$.  One then has that
$$
  \pi (w^*w-1)=0,
  $$
so $w^*w-1\in I$.   Given any $\varepsilon >0$, we then have by hypothesis that there exists a projection $p$ in $I$ (chosen e.g. from
an approximate identity) such that
$$
  \varepsilon >\|(w^*w-1) - p(w^*w-1)\| = $$ $$ =
  \|(1-p)(w^*w-1)\| = \|p^\perp (w^*w-1)\|,
  $$
where  $p^\perp  = 1-p$.
Letting $y=wp^\perp $, observe that
$$
  \pi (y) = \pi (wp^\perp ) = \pi (w-wp)=\pi (w) = u,
  $$
and moreover that
$$
  \|y^*y-p^\perp \| =
  \|p^\perp w^*wp^\perp -p^\perp \| = $$ $$ =
  \|p^\perp (w^*w-1)p^\perp \| \leq
  \|p^\perp (w^*w-1)\|\|p^\perp \| <\varepsilon .
  $$
Since $p^\perp$ is idempotent, anything nearby is approximately idempotent so,  by choosing  $\varepsilon $
sufficiently small, we may assume that
$$
  \|(y^*y)^2 - y^*y\| < \frac14,
  $$
which in turn implies that $\frac12$ cannot be in the spectrum of $y^*y$.
Let
$$
  f:\mathbb R \setminus \left\{\textstyle \frac12\right\} \to \mathbb R
  $$
be the continuous function vanishing identically on $(-\infty ,1/2)$, and such that
$f(t)= t^{-1/2}$, for  $t\in (1/2, \infty )$.  Clearly
$$
  tf(t)^2 = 1_{(\frac12,\infty )},
  $$
which is a real  idempotent function, hence
$$
  f(y^*y)\ y^*y \ f(y^*y)
  $$
is a self-adjoint idempotent element, i.e. a  projection.  Therefore
$$
  v:= y  f(y^*y)
  $$
is a partial isometry thanks to the fact that $v^*v$ is a projection.  To conclude we claim that $\pi (v)=u$, and this is
easy to see since
$$
  \pi (v) = \pi (y)  \pi (f(y^*y)) = \pi (y)  f(\pi (y^*y)) = $$ $$ =
  u  f(u^*u) =   u f(1) = u.
  $$
