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For
$1\le p <\infty $
, we present a reflexive Banach space
$\mathfrak {X}^{(p)}_{\text {awi}}$
, with an unconditional basis, that admits
$\ell _p$
as a unique asymptotic model and does not contain any Asymptotic
$\ell _p$
subspaces. Freeman et al., Trans. AMS.370 (2018), 6933–6953 have shown that whenever a Banach space not containing
$\ell _1$
, in particular a reflexive Banach space, admits
$c_0$
as a unique asymptotic model, then it is Asymptotic
$c_0$
. These results provide a complete answer to a problem posed by Halbeisen and Odell [Isr. J. Math.139 (2004), 253–291] and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of
$\mathfrak {X}^{(p)}_{\text {awi}}$
, we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace.
The purpose of this paper is to quantify the size of the Lebesgue constants
$(\boldsymbol {L}_m)_{m=1}^{\infty }$
associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters
$(\boldsymbol {k}_m)_{m=1}^{\infty }$
determines the growth of
$(\boldsymbol {L}_m)_{m=1}^{\infty }$
. Multiple theoretical applications and computational examples complement our study.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not.2014 (2014), 1289–1311 relating to work of Arveson, Acta Math.118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom.8 (2014), 771–787.
We study the distribution of the roots of a random p-adic polynomial in an algebraic closure of
${\mathbb Q}_p$
. We prove that the mean number of roots generating a fixed finite extension K of
${\mathbb Q}_p$
depends mostly on the discriminant of K, an extension containing fewer roots when it becomes more ramified. We prove further that for any positive integer r, a random p-adic polynomial of sufficiently large degree has about r roots on average in extensions of degree at most r.
Beyond the mean, we also study higher moments and correlations between the number of roots in two given subsets of
${\mathbb Q}_p$
(or, more generally, of a finite extension of
${\mathbb Q}_p$
). In this perspective, we notably establish results highlighting that the roots tend to repel each other and quantify this phenomenon.
We show that
$L_1(L_p) (1 < p < \infty )$
is primary, meaning that whenever
$L_1(L_p) = E\oplus F$
, where E and F are closed subspaces of
$L_1(L_p)$
, then either E or F is isomorphic to
$L_1(L_p)$
. More generally, we show that
$L_1(X)$
is primary for a large class of rearrangement-invariant Banach function spaces.
We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of $L^{p}$ norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it.
The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of
$\mathbb {R}$
accepts an orthogonal basis of exponentials if and only if it tiles
$\mathbb {R}$
by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order
$p^{m}q^{n}$
, when one of the exponents is
$\leq 6$
or when
$p^{m-2}<q^{4}$
, and also prove that a tiling subset of a cyclic group of order
$p_{1}^{m}p_{2}\dotsm p_{n}$
is spectral.
We show that the universal minimal proximal flow and the universal minimal strongly proximal flow of a discrete group can be realized as the Stone spaces of translation-invariant Boolean algebras of subsets of the group satisfying a higher-order notion of syndeticity. We establish algebraic, combinatorial and topological dynamical characterizations of these subsets that we use to obtain new necessary and sufficient conditions for strong amenability and amenability. We also characterize dense orbit sets, answering a question of Glasner, Tsankov, Weiss and Zucker.
In 2016, I solved a problem of de la Harpe from 2006: Is there a nondiscrete C$^{\ast }$-simple group? However the solution was not fully satisfactory, as the C$^{\ast }$-simple groups provided (and their operator algebras) are very close to discrete groups. All previously known examples are of this form. In this article I give yet another construction of nondiscrete C$^{\ast }$-simple groups. The statement in the title then follows. This in particular gives the first examples of nonelementary C$^{\ast }$-simple groups (in Wesolek’s sense).
We obtain a sparse domination principle for an arbitrary family of functions
$f(x,Q)$
, where
$x\in {\mathbb R}^{n}$
and Q is a cube in
${\mathbb R}^{n}$
. When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré–Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vector-valued square functions.
We construct two types of unital separable simple
$C^*$
-algebras:
$A_z^{C_1}$
and
$A_z^{C_2}$
, one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely,
$A_z^{C_i}$
has a unique tracial state,
and
$K_{1}\left (A_z^{C_i}\right )=\{0\}$
(
$i=1,2$
). We show that
$A_z^{C_i}$
(
$i=1,2$
) is essentially tracially in the class of separable
${\mathscr Z}$
-stable
$C^*$
-algebras of nuclear dimension
$1$
.
$A_z^{C_i}$
has stable rank one, strict comparison for positive elements and no
$2$
-quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear)
$C^*$
-algebras which are essentially tracially in the class of simple separable nuclear
${\mathscr Z}$
-stable
$C^*$
-algebras, and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.
We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$, but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.
Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$, $d\geq 2$, equipped with surface measure $\sigma _{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation
$$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$
where $a\in C^\infty (\mathbb {S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb {S}^{d-1})$. We prove that any such solution belongs to the class $C^\infty (\mathbb {S}^{d-1})$. In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb {S}^{d-1}$ are $C^\infty $-smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].
We formulate general conditions which imply that ${\mathcal L}(X,Y)$, the space of operators from a Banach space X to a Banach space Y, has $2^{{\mathfrak {c}}}$ closed ideals, where ${\mathfrak {c}}$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in ${\mathcal L}\left (\ell _p\oplus \ell _q\right )$ is exactly $2^{{\mathfrak {c}}}$ for all $1<p<q<\infty $.
We show that for infinitely many primes p there exist dual functions of order k over ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $-distance by polynomial phase functions of degree $k-1$. This answers in the negative a natural finite-field analogue of a problem of Frantzikinakis on $L_\infty $-approximations of dual functions over ${\mathbb{N}}$ (a.k.a. multiple correlation sequences) by nilsequences.
This article deals with the problem of when, given a collection
$\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in
$\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to
$L_1[0,1]$ factors through Z.
We also prove the following descriptive set theoretical result: Let
$\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if
$\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for
$A \in \mathcal {B}$, the assignment
$A \to A^*$ can be realised by a Borel map
$\mathcal {B}\to \mathcal {L}$.
We establish the global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on products of
$L^p$ spaces by proving endpoint boundedness on suitable product spaces containing combinations of the local Hardy space, the local BMO and the
$L^2$ spaces.
Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the
$L_p$
-norm of the
$\limsup $
of a sequence of operators as a localized version of a
$\ell _\infty /c_0$
-valued
$L_p$
-space. In particular, our main result gives a strong
$L_1$
-estimate for the
$\limsup $
—as opposed to the usual weak
$L_{1,\infty }$
-estimate for the
$\mathop {\mathrm {sup}}\limits $
—with interesting consequences for the free group algebra.
Let
$\mathcal{L} \mathbf{F} _2$
denote the free group algebra with
$2$
generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside
$L_1(\mathcal{L} \mathbf{F} _2)$
for which the free Poisson semigroup converges to the initial data. Currently, the best known result is
$L \log ^2 L(\mathcal{L} \mathbf{F} _2)$
. We improve this result by adding to it the operators in
$L_1(\mathcal{L} \mathbf{F} _2)$
spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative
$\limsup $
together with new transference techniques.
We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak
$(\Phi ,\Phi )$
inequality—as opposed to weak
$(\Phi ,1)$
—for noncommutative multiparametric martingales and
$\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$
. This logarithmic power is an
$\varepsilon $
-perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.
Generalizing von Neumann’s result on type II
$_1$
von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.