For a closed subgroup of a locally compact group the Rieffel induction process gives rise to a C*-correspondence over the C*-algebra of the subgroup. We study the associated Cuntz–Pimsner algebra and show that, by varying the subgroup to be open, compact, or discrete, there are connections with the Exel–Pardo correspondence arising from a cocycle, and also with graph algebras.

The study of existence of a universal C*-completion of the *-algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2($\mathbb{Q}$p), SL2($\mathbb{Z}$p)) does not admit a universal C*-completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell–Rieffel equivalence, and highlighted the role of other C*-completions. In the case of the pair (SLn($\mathbb{Q}$p), SLn($\mathbb{Z}$p)) for n ⩾ 3 we show, invoking property (T) of SLn($\mathbb{Q}$p), that the C*-completion of the L1-Banach algebra and the corner of C*(SLn($\mathbb{Q}$p)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a $\mathfrak{p}$-adic field with a good choice of a maximal compact open subgroup.

We propose a generalization of Exel’s crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler’s theory of Toeplitz and Cuntz–Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz–Pimsner algebras of product systems.

We realize the Hecke C*-algebra $\mathcal{C}_{\mathbb{Q}}$ of Bost and Connes as a direct limit of Hecke C*-algebras which are semigroup crossed products by $\mathbb{N}^F$, for F a finite set of primes. For each approximating Hecke C*-algebra we describe a composition series of ideals. In all cases there is a large type I ideal and a commutative quotient, and the intermediate subquotients are direct sums of simple C*-algebras. We can describe the simple summands as ordinary crossed products by actions of $\mathbb{Z}^S$ for S a finite set of primes. When $\vert S\vert =1$, these actions are odometers and the crossed products are Bunce–Deddens algebras; when $\vert S\vert >1$, the actions are an apparently new class of higher-rank odometer actions, and the crossed products are an apparently new class of classifiable AT-algebras.

We consider a class $(A,\,S,\,\alpha )$ of dynamical systems, where $S$ is an Ore semigroup and $\alpha$ is an action such that each ${{\alpha }_{s}}$ is injective and extendible (i.e. it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of $A$. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group $G$ of $S$ constructed in [15, Proposition 6.1]. It turns out that the full crossed product by this coaction is isomorphic to $A\,{{\rtimes }_{\alpha }}\,S$. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.

A family of Hecke $C^*$ -algebras can be realised as crossed products by semigroups of endomorphisms. It is shown by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.