This is a follow up question on: unitary operators in Hilbert's space are arcwise connected.
I see that Theorem 2.12 of Murphy's book shows that if $\sigma(u) \ne \mathbb{T}$, then there exists $a \in A_{sa}$ such that $u =e^{ia}$ using functional calculus.
I wonder how to continue the argument and show that the unitary operators are arcwise connected as https://math.stackexchange.com/a/3348044/836719 has suggested.
Thank you!
Best Answer
The function $$ \text{lg}:S^1\to[0,2\pi), $$ defined by $\text{lg}(e^{i\theta})=\theta$, for all $\theta \in [0,2\pi)$, is Borel measurable on $\sigma (u)$, so the Borel functional calculus provides a meaning to $\text{lg}(u)$, and one has that $$ e^{i\, \text{lg}(u)}=u. $$