topological stable rank one and AF-algebra construction on Cantor set

A non-example of lifting Answer

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A non-example of lifting

Under some circumstances we will raise a unitary component to a unitary, however it isn’t undoubted for common. Can you give a instance?

Let $I$ breathe a closed model of a $C^*$-algebra $A$ and $u$ breathe a unitary of $A/I$. It just isn’t liftable to a unitary component in $A$, in common.

I used to be eager about this:

If $f in C(mathbb{D})$ is a unitary component and $phi (f) = u$, then $f$ is a steady complex-valued duty on $mathbb{D}$ that doesn’t take the
worth $0$ and coincides with $u$ on $mathbb{T}$. Accordingly, the equation $r(z) = f(z)/|f(z)|$ defines a retraction $r: mathbb{D} rightarrow mathbb{T}$. This is inconceivable, nevertheless, since $mathbb{T}$ just isn’t a retreat of $mathbb{D}$.

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