Lifting a unitary to a partial isometry

Question

What is an example of a unital $C^*$-algebra $A$ and an ideal $I$ such that some unitary element in $A/I$ cannot be lifted to a partial isometry in $A$? Or can it be shown using general properties of unitaries and partial isometries that existence of such a lift forces certain necessary conditions? I read that such a lift does not exist in general but I cannot think of an example.

Answer 1

Suppose that the ideal $I$ is projectionless. If a partial isometry $v\in A$ maps to a unitary, it means that $1-v^*v\in I$, but $1-v^*v$ is a projection, impossible.

This situation can be achieved for instance when $A=C(X)$, $I=C_0(X)$ for some compact subset of $\mathbb C$.

Answer 2

Let $\mathbb T$ be the circle and $\mathbb D$ the disc. The canonical unitary in $C(\mathbb T)$ does not lift to a partial isometry in $C(\mathbb D)$: such a lift would automatically be unitary, violating Brouwer's fixed-point theorem.

The following is true, however (see Lemma 9.2.1 in Rordam-Larsen-Laustsen):

Let $A$ be unital and $u\in A/I$ a unitary. Then $a\oplus 0\in M_2(A/I)$ lifts to a partial isometry in $M_2(A)$.