The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple
$C^*$-algebras arising from free product constructions}, Canad. J.
Math. 50 (1998), 323--341 and
of Theorem 3.1 of K. J. Dykema, Purely infinite simple
$C^*$-algebras arising from free product constructions, II, Math.
Scand. 90 (2002), 73--86 are corrected.
In this paper we analyze states on C*-algebras and
their relationship to filter-like structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the Kadison-Singer conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for q-points in fact also holds for rapid
p-points, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
We prove
restriction and extension of multipliers between
weighted Lebesgue spaces with
two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be
below one.
We also develop some ad-hoc methods which apply to weights
defined by the product of periodic weights with functions of power type.
Our vector-valued approach allow us to extend results
to transference of maximal multipliers and provide transference of Littlewood-Paley inequalities.
We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The non-uniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
This paper provides an addendum and erratum to L. Godinho and
M. E. Sousa-Dias,
"The Fundamental Group of
$S^1$-manifolds". Canad. J. Math. 62(2010), no. 5, 1082--1098.
We define and study the so-called extreme version of the notion of a
projective normed module. The relevant definition takes into account
the exact value of the norm of the module in question, in contrast
with the standard known definition that is formulated in terms of norm
topology.
After the discussion of the case where our normed algebra $A$ is just
$\mathbb{C}$, we concentrate on the case of the next degree of complication,
where $A$ is a sequence algebra, satisfying some natural conditions.
The main results give a full characterization of extremely projective
objects within the subcategory of the category of non-degenerate
normed $A$--modules, consisting of the so-called homogeneous modules.
We consider two cases, `non-complete' and `complete', and the
respective answers turn out to be essentially different.
In particular, all Banach non-degenerate homogeneous modules,
consisting of sequences, are extremely projective within the category
of Banach non-degenerate homogeneous modules. However, neither of
them, provided it is infinite-dimensional, is extremely projective
within the category of all normed non-degenerate homogeneous modules.
On the other hand, submodules of these modules, consisting of finite
sequences, are extremely projective within the latter category.
Permutation products and their various ``fat diagonal'' subspaces are
studied from the topological and geometric point of view. We describe
in detail the stabilizer and orbit stratifications related to the
permutation action, producing a sharp upper bound for its depth and
then paying particular attention to the geometry of the diagonal
stratum. We write down an expression for the fundamental group of any
permutation product of a connected space $X$ having the homotopy type
of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then
prove that the fundamental group of the configuration space of
$n$-points on $X$, of which multiplicities do not exceed $n/2$,
coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving
conditions for when fat diagonal subspaces of manifolds can be
manifolds again. Various examples and homological calculations are
included.
We establish asymptotics for Christoffel functions associated with
multivariate orthogonal polynomials. The underlying measures are assumed to
be regular on a suitable domain - in particular this is true if they are
positive a.e. on a compact set that admits analytic parametrization. As a
consequence, we obtain asymptotics for Christoffel functions for measures on
the ball and simplex, under far more general conditions than previously
known. As another consequence, we establish universality type limits in the
bulk in a variety of settings.
In this paper, we introduce two notions on a surface in a contact
manifold. The first one is called degree of transversality (DOT) which
measures the transversality between the tangent spaces of a surface
and the contact planes. The second quantity, called curvature of
transversality (COT), is designed to give a comparison principle for
DOT along characteristic curves under bounds on COT. In particular,
this gives estimates on lengths of characteristic curves assuming COT
is bounded below by a positive constant.
<p>
We show that surfaces with constant COT exist and we classify all
graphs in the Heisenberg group with vanishing COT. This is
accomplished by showing that the equation for graphs with zero COT can
be decomposed into two first order PDEs, one of which is the backward
invisicid Burgers' equation. Finally we show that the p-minimal graph
equation in the Heisenberg group also has such a
decomposition. Moreover, we can use this decomposition to write down
an explicit formula of a solution near a regular point.
We prove that $r$ independent homogeneous polynomials of the same degree $d$
become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety
whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence
between an algebraic notion (called Weak Lefschetz Property)
and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case,
some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
Darboux Wronskian formulas allow to construct Darboux transformations,
but Laplace transformations, which are Darboux transformations of
order one
cannot be represented this way.
It has been a long standing problem on what are other exceptions. In
our previous work we proved that among transformations of total
order one there are no other exceptions. Here we prove that for
transformations of total order two there are no exceptions at all.
We also obtain a simple explicit invariant description of all possible
Darboux Transformations of total order two.
Meyer sets have a relatively dense set of Bragg peaks and
for this reason they may be considered as basic mathematical
examples of (aperiodic) crystals. In this paper we investigate the
pure point part of the diffraction of Meyer sets in more detail.
The results are of two kinds. First we show that given a Meyer set
and any positive intensity $a$ less than the maximum intensity of its Bragg
peaks, the set of Bragg peaks whose intensity exceeds $a$ is
itself a Meyer set (in the Fourier space). Second we show that if a
Meyer set is modified by addition and removal of points in such a
way that its density is not altered too much (the allowable amount
being given explicitly as a proportion of the original density)
then the newly obtained set still has a relatively dense set of Bragg
peaks.
We analyze the regularity of standing wave solutions
to nonlinear Schrödinger equations of power type on bounded domains,
concentrating on Lipschitz domains. We establish optimal regularity results
in this setting, in Besov spaces and in Hölder spaces.