We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

Suppose that ${\Gamma }^{+ } $ is the positive cone of a totally ordered abelian group $\Gamma $, and $(A, {\Gamma }^{+ } , \alpha )$ is a system consisting of a ${C}^{\ast } $-algebra $A$, an action $\alpha $ of ${\Gamma }^{+ } $ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ is a full corner in the subalgebra of $\L ({\ell }^{2} ({\Gamma }^{+ } , A))$, and that if $\alpha $ is an action by automorphisms of $A$, then it is the isometric crossed product $({B}_{{\Gamma }^{+ } } \otimes A)\hspace{0.167em} {\mathop{\times }\nolimits }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ such that the quotient is the isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $.

We define the class of weakly approximately divisible unital ${C}^{\ast } $-algebras and show that this class is closed under direct sums, direct limits, any tensor product with any ${C}^{\ast } $-algebra, and quotients. A nuclear ${C}^{\ast } $-algebra is weakly approximately divisible if and only if it has no finite-dimensional representations. We also show that Pisier’s similarity degree of a weakly approximately divisible ${C}^{\ast } $-algebra is at most five.

Suppose that $G$ is a second countable, locally compact Hausdorff groupoid with abelian stabiliser subgroups and a Haar system. We provide necessary and sufficient conditions for the groupoid ${C}^{\ast } $-algebra to have Hausdorff spectrum. In particular, we show that the spectrum of ${C}^{\ast } (G)$ is Hausdorff if and only if the stabilisers vary continuously with respect to the Fell topology, the orbit space ${G}^{(0)} / G$ is Hausdorff, and, given convergent sequences ${\chi }_{i} \rightarrow \chi $ and ${\gamma }_{i} \cdot {\chi }_{i} \rightarrow \omega $ in the dual stabiliser groupoid $\widehat{S}$ where the ${\gamma }_{i} \in G$ act via conjugation, if $\chi $ and $\omega $ are elements of the same fibre then $\chi = \omega $.