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
$XX^* + X^*X \subset A$
and a set of certain partial
isometries $s$ in the ``normalizer" of $A$ in $B$,
and similarly for the map $s \mapsto \Ad s$
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 ${\rm II}_1$ 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 ant\'erieur, nous
avions montr\'e que l'induite parabolique (normalis\'ee) d'une
repr\'esentation irr\'eductible cuspidale $\sigma $ d'un
sous-groupe de Levi $M$ d'un groupe $p$-adique contient un
sous-quotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre
\'egal au rang parabolique de $M$. L'objet de cet article est
d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de
Langlands.
Soient $F$ un corps
commutatif localement compact non archim\'edien, $G=\GL
(n,F)$ pour un entier $n\geq 2$, et $\kappa$ un caract\`ere de
$F^\times$ trivial sur $(F^\times)^n$. On prouve une formule pour
les $\kappa$-int\'egrales orbitales r\'eguli\`eres sur $G$
permettant, si $F$ est de caract\'eristique $>0$, de les relever
\`a la caract\'eristique nulle. On en d\'eduit deux r\'esultats
nouveaux en caract\'eristique $>0$\,: le ``lemme fondamental'' pour
l'induction automorphe, et une version simple de la formule des
traces tordue locale d'Arthur reliant $\kappa$-int\'egrales
orbitales elliptiques et caract\`eres $\kappa$-tordus. Cette
formule donne en particulier, pour une s\'erie
$\kappa$-discr\`ete de $G$, les $\kappa$-int\'egrales orbitales
elliptiques d'un pseudo-coefficient comme valeurs du caract\`ere
$\kappa$-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.
