Characterisation of computability of partial functions from infinite words into finite words by functions with prefix-free domain
I guess I have an alternative solution. Let $f :\subseteq \Sigma^{\omega} \to \Sigma^{\omega}$ be computable. Define
$$
 h(u) = v \mbox{ iff }
  f(u\Sigma^{\omega}) = \{v\} \mbox{ with $u$ minimal}
$$
and let $h(u)$ diverge otherwise. Then $\mbox{dom}(h)$ is prefix-free for if $h(u) = v$ then $f(u\Sigma^{\omega}) = \{v\}$ as the machine behaves the same when reading the prefix $u$, producing $v$. Also $h$ is computable, for if $M$ is a machine for $f$ and $u \in \Sigma^{\ast}$, just simulate $M$ on $u0^{\omega}$ and if it produces some output and stops check if the prefix read by $M$ equals $u$; and if not just enter an infinite loop.

Furthermore we have $f = h_{\ast}$. For if $f(p) = u$ then there exists some minimal prefix $w \sqsubseteq p$ with $f(w\Sigma^{\omega}) = \{u\}$, hence $h_{\ast}(p) = h(w) = u$. For the other direction suppose $h_{\ast}(p) = u$, then $w \sqsubseteq p$ with $h(w) = u$, i.e. $f(w\Sigma^{\omega}) = \{u\}$, in particular $f(p) = u$.