We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting 
$2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia’s CCR conjecture on trees for 
$(d_0,d_1)$-semi-regular trees such that 
$d_0,d_1in Theta $, where 
$Theta $ is an asymptotically dense set of positive integers.
We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles.
In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure.
The complex algebra of an inverse semigroup with finitely many idempotents in each 
$mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using 
$C^{*}$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupoids. We construct here countably many nonisomorphic finitely presented inverse semigroups with finitely many idempotents in each 
$mathcal D$-class and non-Hausdorff universal groupoids. At this time, there is not a clear 
$C^{*}$-algebraic technique to prove these inverse semigroups have stably finite complex algebras.
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