In his C$^{*}$-algebra book, Davidson writes the following:
I am not quite clear what he means by $\mathcal{B}(\mathcal{H})$ here.
Is $\mathcal{H}$ a fixed Hilbert space or are we considering $(U_{\alpha},V_{\alpha})$ sitting in $\mathcal{B}(\mathcal{H})$ for any $\mathcal{H}$?
If it's the latter case, I am confused about how $\widetilde{U}$ and $\widetilde{V}$ are defined since the collection of all Hilbert spaces don't form a set.
However, if $\mathcal{H}$ is a fixed Hilbert space, then I have the following problem. On the next page, two arbitrary unitaries $U$ and $V$satisfying $(\dagger)$ are considered, along with an irreducible representation $\pi$ of $C^{*}(U,V)$. Davidson argues that $U'=\pi(U)$ and $V'=\pi(V)$ form an irreducible pair satisfying $(\dagger)$ (which I totally understand). But then he concludes that $\|U'\|\leq \|\widetilde{U}\|$ and $\|V'\|\leq \|\widetilde{V}\|$ I assume by concluding that $(U',V')$ is one of the $(U_{\alpha},V_{\alpha})$ used in the definition of $\widetilde{U}$ and $\widetilde{V}$. How is this possible given that, in this case, $U'$ and $V'$ may not sit on the fixed Hilbert space $\mathcal{H}$?
Thank you for your time.
A C$^*$-algebra of the form $C^*(U,V)$ is always separable, and so it will always admit a faithful representation into $B(H)$ with $H$ separable (see I.9.12 in Davidson). In particular, it can faithfully represented into $B(L^2(\mathbb R\setminus \mathbb Z))$, which is the $B(H)$ Davidson is using. So the pairs are taken from within a single $B(H)$.
As for your second question, if $A=C^*(U',V')$, take a faithful representation $\pi:A\to B(H)$. Then $(\pi(U'),\pi(V')$ are in the list, and so $$ \|U'\|=\|\pi(U')\|\leq\|\tilde U\|, $$ and similarly for $V'$.