  The `Bohrification" program in the foundations of quantum mechanics
implements Bohr's doctrine of classical concepts through an interplay between
commutative and non-commutative operator algebras. Following a brief conceptual
and mathematical review of this program, we focus on one half of it, called
"exact" Bohrification, where a (typically noncommutative) unital C*-algebra A
is studied through its commutative unital C*-subalgebras, organized into a
poset C(A). This poset turns out to be a rich invariant of A. To set the stage,
we first give a general review of symmetries in elementary quantum mechanics
(i.e., on Hilbert space) as well as in algebraic quantum theory, incorporating
C(A) as a new kid in town. We then give a detailed proof of a deep result due
to Hamhalter (2011), according to which C(A) determines A as a Jordan algebra
(at least for a large class of C*-algebras). As a corollary, we prove a new
Wigner-type theorem to the effect that order isomorphisms of C(B(H)) are (anti)
unitarily implemented. We also show how C(A) is related to the orthomodular
poset P(A) of projections in A. These results indicate that C(A) is a serious
player in C*-algebras and quantum theory.
