(1) The positive quotient and polar decomposition. How does polar decomposition generalize to all unital $C^*$-algebras, and not just the supported ones? This question may be answered in terms of a canonical correspondence between wide étale subcategories of $\mathcal{M}$ and the quotients in $\mathscr{M}$ of the representable presheaf associated with the unit of $\mathcal{A}$.
(2) The Gelfand spectrum of a commutative $C^*$-algebra: 1-dimensional representations. How this is related to $\mathcal{M}$ follows a pattern similar to the Zariski spectrum of a commutative ring.
(3) GNS-representation theory (in any dimension), and the spectrum of an arbitrary
$C^*$-algebra. One of our tools is what we shall call a seminormed $\mathcal{A}$-module,
by which we mean a right $\mathcal{A}$-module $\mathcal{V}$ that carries a seminorm such that
\[
\forall v \in \mathcal{V}, \; \forall U \in \mathcal{A}, \quad \|vU\| \leq \|v\| \, \|U\| .
\]
The functional dual $B(\mathcal{A})$, consisting of bounded $\mathbb{C}$-linear maps
$\tau : \mathcal{A} \to \mathbb{C}$, is an important $\mathcal{A}$-module especially
for GNS-theory. Its right action is defined by
\[
\tau U(X) = \tau(XU^*), \quad \tau \in B(\mathcal{A}).
\]
Moreover, this action satisfies $\|\tau U\| \leq \|\tau\| \, \|U\|$, so $B(\mathcal{A})$
is a normed $\mathcal{A}$-module. $B(\mathcal{A})$ classifies functionals in the sense that
for any seminormed $\mathcal{A}$-module $\mathcal{V}$ there is a natural isomorphism
\[
B(\mathcal{V}) \cong \mathcal{A}\text{-Bdd}(\mathcal{V}, B(\mathcal{A})).
\]
Therefore, the presheaf associated with $B(\mathcal{A})$ is given by
\[
\widehat{B(\mathcal{A})}(R) = \mathcal{A}\text{-Bdd}(R\mathcal{A}, B(\mathcal{A}))
\;\cong\; B(R\mathcal{A}).
\]
From this point of view, $\widehat{B(\mathcal{A})}$ is the presheaf of functional germs
on $\mathcal{A}$, suggesting that $\widehat{B(\mathcal{A})}$ may be interpreted as the
complex numbers object of the topos $\mathscr{M}$. In any case, $\widehat{B(\mathcal{A})}$
is a ring object with a conjugation operation (such that
$\overline{\tau}(SX) = \overline{\tau(SX)}$) internal to $\mathscr{M}$.