The Components of $\mathcal{SF}$

Let $G$ be any topological group and let $G_0$ be the component of the identity. Show that $G_0$ is a normal subgroup of $G$.

What are the components of the set of invertible elements in $C(\partial \mathbb {D})$?

If $S =$ the unilateral shift, what are the components of the set of invertible elements of $C^*(S)$?

If $U$ and $V$ are isometries, show that there is a path consisting entirely of isometries connecting $U$ to $V$ if and only if $\dim (\mathrm{ran}\ U)^\perp = \dim (\mathrm{ran}\ V)^\perp$.

Find the components of the set of partial isometries.