The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.

For monotone complete ${{C}^{*}}$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed ${{C}^{*}}$-subalgebra, the relation $X=AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX+XA\subset X$ and $X{{X}^{*}}+{{X}^{*}}X\subset A$ and a set of certain partial isometries $s$ in the “normalizer” of $A$ in $B$, and similarly for the map $s\mapsto \text{Ad }s$ between the latter set and a set of certain “partial $*$-automorphisms” of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type $\text{I}{{\text{I}}_{1}}$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the ${{C}^{*}}$-algebra version of these results.

Dans un travail antérieur, nous avions montré que l’induite parabolique (normalisée) d’une représentation irréductible cuspidale $\sigma $ d’un sous-groupe de Levi $M$ d’un groupe $p$-adique contient un sous-quotient de carré intégrable, si et seulement si la fonction $\mu $ de Harish-Chandra a un pôle en $\sigma $ d’ordre égal au rang parabolique de $M$. L’objet de cet article est d’interpréter ce résultat en termes de fonctorialité de Langlands.

Soient $F$ un corps commutatif localement compact non archimédien, $G=GL(n,F)$ pour un entier $n\ge 2$, et $\kappa$ un caractère de ${{F}^{\times }}$ trivial sur ${{\left( {{F}^{\times }} \right)}^{n}}$. On prouve une formule pour les $\kappa$-intégrales orbitales régulières sur $G$ permettant, si $F$ est de caractéristique $>0$, de les relever à la caractéristique nulle. On en déduit deux résultats nouveaux en caractéristique $>0$ : le “lemme fondamental” pour l’induction automorphe, et une version simple de la formule des traces tordue locale d’Arthur reliant $\kappa$-intégrales orbitales elliptiques et caractères $\kappa$-tordus. Cette formule donne en particulier, pour une série $\kappa$-discrète de $G$, les $\kappa$-intégrales orbitales elliptiques d’un pseudo-coefficient comme valeurs du caractère $\kappa$-tordu.

We produce a complete description of the lattice of gauge-invariant ideals in ${{C}^{*}}(\Lambda )$ for a finitely aligned $k$-graph $\Lambda $. We provide a condition on $\Lambda $ under which every ideal is gauge-invariant. We give conditions on $\Lambda $ under which ${{C}^{*}}(\Lambda )$ satisfies the hypotheses of the Kirchberg–Phillips classification theorem.

We give the general structure of complex (resp., real) $G$-graded contractions of Lie algebras where $G$ is an arbitrary finite Abelian group. For this purpose, we introduce a number of concepts, such as pseudobasis, higher-order identities, and sign invariants. We characterize the equivalence classes of $G$-graded contractions by showing that our set of invariants (support, higher-order identities, and sign invariants) is complete, which yields a classification.