 # 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.

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 $$mathbb{D}$$.