A stably Gelfand quantale is an involutive quantale $Q$ that satisfies \[aa^*a\le a\quad \Longrightarrow\quad aa^*a=a\] for all $a\in Q$. Examples include the quantale $\opens(G)$ of an \'etale groupoid $G$, and the quantale $\Max A$ of closed linear subspaces of a C*-algebra $A$. To each projection $b$ of a stably Gelfand quantale $Q$ is associated a complete and infinitely distributive inverse semigroup $\mathfrak B\subset Q$ consisting of all the elements $a\in Q$ such that

\begin{eqnarray*}

a^*a&\le& b\;,\\

aa^*&\le& b\;,\\

ab&\le& a\;,\\

ba&\le& a\;.

\end{eqnarray*}

Consequently, we also obtain a localic \'etale groupoid $\mathcal B$ via the quantale $\opens(\mathcal B)=\lcc(\mathfrak B)$. The inclusion $\mathfrak B\to Q$ extends to a quantale homomorphism $\mathfrak b^*:\opens(\mathcal B)\to Q$, which, by analogy with the corresponding definition for locales, we regard as the ``inverse image homomorphism'' of a ``continuous map'' of quantales

\[

\mathfrak b:Q\to\opens(\mathcal B)\;.

\]

If $Q=\Max A$ for a C*-algebra $A$, a projection is the same as a sub-C*-algebra $B$, and we view the pair $(A,\mathfrak b)$ as a ``C*-algebraic bundle'' over $\mathcal B$. Conversely, any such bundle $(A,p:\Max A\to\opens(G))$ yields a sub-C*-algebra $p^*(e)\subset\Max A$, and thus we obtain a general constructive framework that mimics the interplay between Cartan sub-C*-algebras and Fell bundles on \'etale groupoids and inverse semigroups. Whereas the latter hinges on the existence of faithful conditional expectations and additional conditions such as commutativity and maximality of subalgebras, the language of quantales and inverse semigroups suggests other natural properties of a sub-C*-algebra $B$. Comparing both sets of properties may be interesting in its own right. For instance, we say that $B$ is localic if $\mathfrak b$ is a surjection; and that $B$ is open if $\mathfrak b$ is an open map in a sense that generalizes open maps of locales and such that surjections are stable under pullbacks. As an example, Cartan subalgebras in the sense of Renault are localic, which can be proved by showing that any Fell line bundle on a locally compact Hausdorff groupoid $G$ yields a surjection $p:\Max C_r^*(\pi)\to\topology(G))$. It is not clear whether $p$ is open in general, but in some examples it is, for instance if $G$ is compact.

In this talk I will describe this theory to some extent, along with mentioning open questions and related ongoing or future research.