Let $\Omega\subset \Omega^d$, $d\in \{2,3\}$, be a bounded $d$-polyhedron with $n$ faces. Denote the faces of $\partial\Omega$ as $\{e_i\}_{i=1}^n$. Let $u\in H^{1/2}(\partial\Omega)$ Taking the definition of the $H^{1/2}$ norm as
$$\| v\|_{H^{1/2}(\partial\Omega)} = \inf_{p\in H^1(\Omega)} \|p\|_{H^1(\Omega)}, $$ where in the infinum we require $p\big|_{x\in \partial\Omega} = v$.
How does one extend this definition to $\|v\|_{H^{1/2}(e_i)}$? A seemingly natural way would be to define a new function that is the value of $v$ on $e_i$ and the value of zero everywhere else. And define the norm of $v$ on $e_i$ to be the norm of this new function over all of $\partial\Omega$. Unfortunately we have no guarantees that this new function is in $H^{1/2}(\partial\Omega)$.
So my question is, how do we define the $H^{1/2}(e_i)$ norm. I know that we can use the Fourier Transform definition of this norm but I am wondering if there is way analogous to the above.
Not really. If a function has value $v$ on $e_i$ that is non-zero on any compactly supported subset of $e_i$, and $0$ elsewhere, it does not have $H^{1/2}$ regularity on the whole boundary: $$ \int_{\partial \Omega}\int_{\partial \Omega}\frac{|p(x) - p(y)|^2}{|x-y|^2} dS(x) dS(y) $$ can be unbounded (consider $v=1$ on $e_i$, zero elsewhere on boundary, then the sharp gradient is unbounded near $e_i$'s co-boundary).
Now, following the quotient norm tradition (use the infimum of the extension to define the boundary norm), then there does not exist a $p\in H^1(\Omega)$ such that the following extension is true, due to the fact that the Dirichlet data is not in $H^{1/2}(\partial\Omega)$: $$ -\Delta p = 0, \quad p = v \text{ on } e_i, \quad p = 0 \text{ on }\partial{\Omega}\backslash e_i. $$
However, the following problem does have a unique solution $p\in H^1(\Omega)$: $$ -\Delta p' = 0, \quad p' = v \text{ on } e_i, \quad \frac{\partial p'}{\partial n} = 0 \text{ on }\partial{\Omega}\backslash e_i. \tag{$\star$} $$ Also you have $\|v\|_{H^{1/2}(e_i)} \leq \|\operatorname{trace}(p')\|_{H^{1/2}(\partial \Omega)} \leq C \|p'\|_{H^1(\Omega)}$. By the a priori estimate for problem $(\star)$, $\|p'\|_{H^1(\Omega)} \leq \|v\|_{H^{1/2}(e_i)} $. As a result, using this extension, you can define a quotient norm in this way: $$ \|v\|_{H^{1/2}(e_i)} := \inf_{p\in H^1} \|p\|_{H^1(\Omega)}, $$ where $p = v$ on $e_i$, and $\frac{\partial p}{\partial n} = 0 $ on $\partial{\Omega}\backslash e_i$.