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I think this is best viewed as combination of two facts. The first is that unitary representations of $G$ are interchangeable with representations of $C^*(G)$ (proved using $L^1(G)$). This can be found in e.g. Dixmier 13.9.3. The second is the extended version of the GNS correspondence for C*-algebras that applies also to non-positive linear functionals. I don't have a reference for this, but it can be proved using the polar decomposition of a linear functional (in the enveloping W*-algebra). There may be an easier way.
You write that tauto cannot prove ¬¬A -> A => (A v ¬A). This is because it is false in intuitionistic logic. The open sets of topological spaces give an interpretation of intuitionistic logic, where or and and are given by union and intersection, and ¬ is interpreted as the interior of the complement. Then (0,1) in the reals gives a counterexample, ¬¬(0,1) = (0,1) but (0,1) v ¬(0,1) is not equal to R. In fact for topological spaces the statement you suggested is equivalent to "all regular open sets are clopen".
The number of degrees of freedom is the dimension of the classical configuration space (in the sense of classical mechanics). Harmonic oscillators are simple examples where the number of degrees of finite (it can even be 1) and the dimension of the quantum Hilbert space is infinite.
As Michael Greinecker says, the closed convex sets differ in general. We could also consider the kernel of a continuous linear functional. Weak-* closed convex and closed convex are the same if and only if the space is reflexive, i.e. the weak and weak-* topologies coincide.
Could you clarify the question? Were you asking whether: a) The class of all isomorphism classes is a set, or b) Each isomorphism class is a set? Denis Nardin's answer is appropriate for (a), but not for (b). The full subcategory of the category of sets on the singletons is an example where (a) is true but not (b). The full subcategory of the category of sets on the cardinals is an example where (b) is true but not (a).
Can you tell us which books? This would help us to decide which of the answers given is right, though my previous experience of category theory in computer science suggests that it is Andreas Blass.
It is important to note that "ignore" and "expose" do not have the same meaning as they do in English. In particular, I find expose quite difficult to translate. Also, "croissant" means "increasing".
