3,854 research outputs found
Analytic vectors in continuous p-adic representations
Given a compact p-adic Lie group G over a finite unramified extension L/Q_p
let G_0 be the product over all Galois conjugates of G. We construct an exact
and faithful functor from admissible G-Banach space representations to
admissible locally L-analytic G_0-representations that coincides with passage
to analytic vectors in case L=Q_p. On the other hand, we study the functor
"passage to analytic vectors" and its derived functors over general basefields.
As an application we determine the higher analytic vectors in certain locally
analytic induced representations.Comment: Final version (appeared in Comp. Math. 2009). Exposition shortened.
Minor items correcte
The Riesz representation theorem and weak compactness of semimartingales
We show that the sequential closure of a family of probability measures on
the canonical space of c{\`a}dl{\`a}g paths satisfying Stricker's uniform
tightness condition is a weak compact set of semimartingale measures in
the pairing of the Riesz representation theorem under topological assumptions
on the path space. Similar results are obtained for quasi- and supermartingales
under analogous conditions. In particular, we give a full characterization of
the strongest topology on the Skorokhod space for which these results are true.Comment: V1-V6 differ substantially from v7-v8 in exposition v9 adds lemma
3.5, example 3.9, remark 5.6, proposition 5.7 (i) and some other minor
remark
Algebras of p-adic distributions and admissible representations
Let G be a compact, locally L-analytic group, where L is a finite extension
of Qp. Let K be a discretely valued extension field of L. We study the algebra
D(G,K) of K-valued locally analytic distributions on G, and apply our results
to the locally analytic representation theory of G in vector spaces over K. Our
objective is to lay a useful and powerful foundation for the further study of
such representations.
We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the
ring of functions on a rigid Stein space, and that (at least when G is
Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]],
where Zp[[G]] is the completed group ring of G. We use this point of view to
describe an abelian subcategory of D(G,K) modules that we call coadmissible.
We say that a locally analytic representation V of G is admissible if its
strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is
admissible if its strong dual is coadmissible as D(H,K) module for some compact
open subgroup H. In this way we obtain an abelian category of admissible
locally analytic representations. These methods allow us to answer a number of
questions raised in our earlier papers on p-adic representations; for example
we show the existence of analytic vectors in the admissible Banach space
representations of G that we studied in "Banach space representations ...",
Israel J. Math. 127, 359-380 (2002).
Finally we construct a dimension theory for D(G,K), which behaves for
coadmissible modules like a regular ring, and show that smooth admissible
representations are zero dimensional
Uniform topologies on types
We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet 1989). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris 2006). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.Rationalizability, incomplete information, higher-order beliefs, strategic topology, electronic mail game
Limit operators, collective compactness, and the spectral theory of infinite matrices
In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on
Spectral synthesis for Banach Algebras II
This paper continues the study of spectral synthesis and the topologies tau-infinity and tau-r on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C*-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of tau_infinity. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of tau_r
On duality theory and pseudodifferential techniques for Colombeau algebras: generalized delta functionals, kernels and wave front sets
Summarizing basic facts from abstract topological modules over Colombeau
generalized complex numbers we discuss duality of Colombeau algebras. In
particular, we focus on generalized delta functionals and operator kernels as
elements of dual spaces. A large class of examples is provided by
pseudodifferential operators acting on Colombeau algebras. By a refinement of
symbol calculus we review a new characterization of the wave front set for
generalized functions with applications to microlocal analysis
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