  This thesis investigates parametrized quantum spin systems in the
thermodynamic limit from a $C^*$-algebraic point of view. Our main physical
result is the construction of a phase invariant for one-dimensional quantum
spin chains parametrized by a topological space $X$. This invariant is
constructed using $C^*$-algebraic techniques and takes values in degree one
\v{C}ech cohomology $H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))$, where
$\mathbb{P}\mathrm{U}(\mathscr{H})$ is the projective unitary group of an
infinite-dimensional Hilbert space $\mathscr{H}$, endowed with the strong
operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may
equivalently view this as an element of $H^3(X;\mathbb{Z})$. An exactly
solvable model of a one-dimensional spin system parametrized by the 3-sphere $X
= \mathbb{S}^3$ is presented and it is shown that its invariant is nontrivial
[WQB+22].
  We also prove several mathematical results on topological aspects of the pure
state space $\mathscr{P}(\mathfrak{A})$ of a $C^*$-algebra $\mathfrak{A}$. We
prove that $\mathscr{P}(\mathfrak{A})$, endowed with the weak* topology, has
trivial fundamental group for every UHF algebra $\mathfrak{A}$ [BHM+23]. We
review the results of [SMQ+22] that show how the outputs of the GNS
representation and Kadison transitivity theorem can be understood to depend
continuously on their inputs. These results are expanded upon with the
construction of a distinguished \v{C}ech class in
$H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))$ that generalizes the principal
$\mathrm{U}(1)$-bundle $\mathbb{S} \mathscr{H} \rightarrow \mathbb{P}
\mathscr{H}$ in a representation independent way, where $\mathbb{S}
\mathscr{H}$ and $\mathbb{P} \mathscr{H}$ are the unit sphere and projective
Hilbert space of $\mathscr{H}$. Finally, we prove a selection theorem in the
vein of [SMQ+22] for the weak* topology on $\mathscr{P}(\mathfrak{A})$.
