116,345 research outputs found
Remarks on the non-commutative Khintchine inequalities for
We show that the validity of the non-commutative Khintchine inequality for
some with implies its validity (with another constant) for all
. We prove this for the inequality involving the Rademacher
functions, but also for more general "lacunary" sequences, or even
non-commutative analogues of the Rademacher functions. For instance, we may
apply it to the "Z(2)-sequences" previously considered by Harcharras. The
result appears to be new in that case. It implies that the space
contains (as an operator space) a large subspace uniformly isomorphic (as an
operator space) to with . This naturally raises
several interesting questions concerning the best possible such .
Unfortunately we cannot settle the validity of the non-commutative Khintchine
inequality for but we can prove several would be corollaries. For
instance, given an infinite scalar matrix , we give a necessary and
sufficient condition for to be in the Schatten class for
almost all (independent) choices of signs . We also characterize the
bounded Schur multipliers from to . The latter two characterizations
extend to results already known for . In addition, we
observe that the hypercontractive inequalities, proved by Carlen and Lieb for
the Fermionic case, remain valid for operator space valued functions, and hence
the Kahane inequalities are valid in this setting.Comment: Some more minor correction
Subgaussian sequences in probability and Fourier analysis
This is a review on subgaussian sequences of random variables, prepared for
the Mediterranean Institute for the Mathematical Sciences (MIMS). We first
describe the main examples of such sequences. Then we focus on examples coming
from the harmonic analysis of Fourier series and we describe the connection of
subgaussian sequences of characters on the unidimensional torus (or any compact
Abelian group) with Sidon sets. We explain the main combinatorial open problem
concerning such subgaussian sequences. We present the answer to the analogous
question for subgaussian bounded mean oscillation (BMO) sequences on the unit
circle. Lastly, we describe several very recent results that provide a
generalization of the preceding ones when the trigonometric system (or its
analogue on a compact Abelian group) is replaced by an arbitrary orthonormal
system bounded in .Comment: We added a short section on subgaussian BMO sequence
Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets
We present a proof of Rider's unpublished result that the union of two Sidon
sets in the dual of a non-commutative compact group is Sidon, and that randomly
Sidon sets are Sidon. Most likely this proof is essentially the one announced
by Rider and communicated in a letter to the author around 1979 (lost by him
since then). The key fact is a spectral gap property with respect to certain
representations of the unitary groups that holds uniformly over . The
proof crucially uses Weyl's character formulae. We survey the results that we
obtained 30 years ago using Rider's unpublished results. Using a recent
different approach valid for certain orthonormal systems of matrix valued
functions, we give a new proof of the spectral gap property that is required to
show that the union of two Sidon sets is Sidon. The latter proof yields a
rather good quantitative estimate. Several related results are discussed with
possible applications to random matrix theory.Comment: v2: minor corrections, v3 more minor corrections v4) minor
corrections, last section removed to be included in another paper in
preparation with E. Breuillard v5) more minor corrections + two references
added. The paper will appear in a volume dedicated to the memory of V. P.
Havi
Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems
We describe the first results of a project of analyzing in which theories
formal proofs can be ex- pressed. We use this analysis as the basis of
interoperability between proof systems.Comment: In Proceedings PxTP 2017, arXiv:1712.0089
Rigidity and cohomology of hyperbolic manifolds
When X=\Gamma\backslash \H^n is a real hyperbolic manifold, it is already
known that if the critical exponent is small enough then some cohomology spaces
and some spaces of harmonic forms vanish. In this paper, we show rigidity
results in the borderline case of these vanishing results
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