Set-theoretic questions about the definitions of crossed-product $ C^{*} $-algebras and group $ C^{*} $-algebras. In his book Crossed Products of $ C^{*} $-Algebras, Dana P. Williams defines the crossed product of a $ C^{*} $-algebra $ A $ by a locally compact group $ G $ as the completion of $ {C_{c}}(G,A) $ with respect to the norm $ \| \cdot \| $ defined by
$$
\| f \| \stackrel{\text{df}}{=}
\sup(\{
\| (\pi \rtimes U)(f) \| \mid
\text{$ (\pi,U) $ is a covariant representation of $ (G,A,\alpha) $}
\}). \qquad (1)
$$
It is said, on page 52 of the book, that the the collection of values in $ (1) $ is a subclass of the set $ \Bbb{R} $ of real numbers, that the separation axioms of set theory guarantee that a subclass of a set is yet a set, and that we are taking the supremum of a bounded set of real numbers.
In the book Morita Equivalence and Continuous-Trace $ C^{*} $-Algebras by Iain Raeburn and Dana P. Williams, the group $ C^{*} $-algebra of a locally compact group $ G $ is defined as the completion of $ {C_{c}}(G) $ with respect to the norm $ \| \cdot \| $ defined by
$$
\| f \| \stackrel{\text{df}}{=}
\sup(\{
\| \pi(f) \| \mid
\text{$ \pi $ is a cyclic norm-decreasing representation of $ {C_{c}}(G) $}
\}). \qquad(2)
$$
It is said, on page 281 of this book, that “one takes cyclic representations in the definition of $ \| \cdot \| $ merely to guarantee that we are taking the supremum over a set.”
My question is why the collection in $ (1) $ is a set (I only know a little about set theory and cannot understand the explanation), but in $ (2) $, we need to take cyclic representations to guarantee that we are taking the supremum over a set.
Thanks!
 A: The Axiom Schema of Separation tells us that if $ A $ is a set and $ \varphi $ is a formula with a free variable, then the collection
$$
\{ x \mid x \in A ~ \text{and} ~ \varphi[x] \}
$$
is a set. Now, fix both $ (G,A,\alpha) $ and $ f \in {C_{c}}(G,A) $, and let $ \varphi[r] $ denote the sentence

There exists a covariant representation $ (\pi,U) $ of $ (G,A,\alpha) $ such that $ r = \| (\pi \rtimes U)(f) \| $.

As $ \Bbb{R} $ is a set, it follows from Separation that the collection
$$
\{ r \mid r \in \Bbb{R} ~ \text{and} ~ \varphi[r] \}
$$
is a set, which is precisely the collection labeled $ (1) $ in your post. It is a non-empty and bounded subset of $ \Bbb{R} $, so its supremum is a real number. Its boundedness is a consequence of the fact that the integrated form $ \pi \rtimes U $ of a covariant representation $ (\pi,U) $ is norm-decreasing.
Truthfully speaking, whether or not we take only cyclic norm-decreasing representations in $ (2) $ does not affect its status as a set. By Separation, even if we drop the ‘cyclic’ condition, it is still a set. We must keep the ‘norm-decreasing’ condition, however, if we are to have a bounded subset of $ \Bbb{R} $.
My guess is that since Williams wrote Morita Equivalence and Continuous-Trace $ C^{*} $-Algebras before Crossed Products of $ C^{*} $-Algebras, he must have realized this point only when writing the second book.
