Connected components of invertibles, unitaries and isometries

Prove that if $U$ is any unitary operator on $\mathcal{H}$, then there is a continuous function $u: [0, 1] \to \mathcal{B}(\mathcal{H})$ such that $u(t)$ is unitary for all $t$, $u(0) = U$, and $u(1) = 1$.

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.

Let $C_{R}$ be the algebra of all real continuous functions on $[0,1]$, with the supremum norm. This satisfies all requirements of a Banach algebra, except that the scalars are now real.

Suppose $A = C(T)$, the algebra of all continuous complex functions on the unit circle $T$, with the supremum norm. Show that two invertible members of $C(T)$ are in the same coset of $G_{1}$ if and only if they are homotopic mappings of $T$ into the set of all nonzero complex numbers. Deduce from this that $G / G_{1}$ is isomorphic to the additive group of the integers. (The notation is as in Theorem 10.34).

Suppose $A = M(R)$, the convolution algebra of all complex Borel measures on the real line; see (e) of Example 10.3. Supply the details in the following proof that $G / G_{1}$ is uncountable: If $\alpha \in R$, let $\delta_{\alpha }$ be the unit mass concentrated at $\alpha$. Assume $\delta_{\alpha } \in G_{1}$. Then $\delta_{\alpha } = \exp \left(\mu_{\alpha }\right)$ for some $\mu_{\alpha } \in M(R)$; hence, for $-\infty < t < \infty$,

where $k$ is an integer. Since $\hat{\mu }_{\alpha }$ is a bounded function, $\alpha = 0$. Thus $\delta_{0}$ is the only $\delta_{\alpha }$ in $G_{1}$. No coset of $G_{1}$ in $G$ contains therefore more than one $\delta_{\alpha }$.