DOCSLIB.ORG

Nonarchimedean Functional Analysis Peter Schneider

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nonarchimedean Functional Analysis Peter Schneider Nonarchimedean Functional Analysis Peter Schneider Version: 25.10.2005 1 This book grew out of a course which I gave during the winter term 1997/98 at the Universit¨at M¨unster. The course covered the material which here is presented in the first three chapters. The fourth more advanced chapter was added to give the reader a rather complete tour through all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields. There is one serious restriction, though, which seemed inevitable to me in the interest of a clear presentation. In its deeper aspects the theory depends very much on the field being spherically complete or not. To give a drastic example, if the field is not spherically complete then there exist nonzero locally convex vector spaces which do not have a single nonzero continuous linear form. Although much progress has been made to overcome this problem a really nice and complete theory which to a large extent is analogous to classical functional analysis can only exist over spherically complete fields. I therefore allowed myself to restrict to this case whenever a conceptual clarity resulted. Although I hope that this text will also be useful to the experts as a reference my own motivation for giving that course and writing this book was different. I had the reader in mind who wants to use locally convex vector spaces in the applications and needs a text to quickly grasp this theory. There are several areas, mostly in number theory like p-adic modular forms and deformations of Galois representations and in representation theory of p-adic reductive groups, in which one can observe an increasing interest in methods from nonarchimedean functional analysis. By the way, discretely valued fields like p-adic number fields as they occur in these applications are spherically complete. This is a textbook which is self-contained in the sense that it requires only some basic knowledge in linear algebra and point set topology. Everything presented is well known, nothing is new or original. Some of the material in the last chapter appears in print for the first time, though. In the references I have listed all the sources I have drawn upon. At the same time this list shows to the reader who the protagonists are in this area of mathematics. I certainly do not belong to this group. M¨unster, May 2001 Peter Schneider 2 List of contents Chap. I: Foundations §1 Nonarchimedean fields §2 Seminorms §3 Normed vector spaces §4 Locally convex vector spaces §5 Constructions and examples §6 Spaces of continuous linear maps §7 Completeness §8 Fr´echet spaces §9 The dual space Chap. II: The structure of Banach spaces §10 Structure theorems §11 Non-reflexivity Chap. III: Duality theory §12 c-compact and compactoid submodules §13 Polarity §14 Admissible topologies §15 Reflexivity §16 Compact limits Chap. IV: Nuclear maps and spaces §17 Topological tensor products §18 Completely continuous maps §19 Nuclear spaces §20 Nuclear maps §21 Traces §22 Fredholm theory References Index, Notations 3 Chap. I: Foundations In this chapter we introduce the basic notions and constructions of nonar- chimedean functional analysis. We begin in §1 with a brief but self-contained review of nonarchimedean fields. The main objective of functional analysis is the investigation of a certain class of topological vector spaces over a fixed nonar- chimedean field K. This is the class of locally convex vector spaces. The more traditional analytic point of view characterizes locally convex topologies as those vector space topologies which can be defined by a family of (nonarchimedean) seminorms. But the presence of the ring of integers o inside the field K allows for an equivalent algebraic point of view. A locally convex topology on a K-vector space V is a vector space topology defined by a class of o-submodules of V which are required to generate V as a vector space. In §§2 and 4 we thoroughly discuss these two concepts and their equivalence. Throughout the book we usually will present the theory from both angles. But sometimes there will be a certain bias towards the algebraic point of view. The most basic methods to actually construct locally convex vector spaces along with concrete examples are treated in §§3 and 5. In §6 we explain how the notion of a bounded subset leads to a systematic method to equip the vector space of continuous linear maps between two given locally convex vector spaces with a natural class of locally convex topologies. The two most important ones among them are the weak and the strong topology. The important concepts of completeness and quasi-completeness are discussed in §7. The construction of the completion of a locally convex vector space is one of the places where we find an algebraic treatment preferable since conceptually simpler. Banach spaces as already introduced in §3 are complete. They are included in the very important class of Fr´echet spaces. These are the complete locally convex vector spaces whose topology is metrizable. Their importance partly derives from the validity of the closed graph and open mapping theorems for linear maps between Fr´echet spaces. These basic results are established in §8 using Baire category theory. In the final §9 of this chapter we begin the investigation of the continuous linear dual of a locally convex vector space. Provided the field K is spherically complete we establish the Hahn-Banach theorem about the existence of continuous linear forms. This is then applied to obtain the first properties of the duality maps into the various forms of the linear bidual. In this section we encounter for the first time the phenomenon in nonarchimedean functional analysis that crucial aspects of the theory depend on special properties of the nonarchimedean field K. The ultimate reason for this difficulty is that K need not to be locally compact. A satisfactory substitute for compact subsets in locally convex K- vector spaces only exists if the field K is spherically complete. This will be discussed systematically in §12 of the third chapter. §1 Nonarchimedean fields 4 Let K be a field. A nonarchimedean absolute value on K is a function | | : K −→ IR such that, for any a,b ∈ K we have (i) |a| ≥ 0, (ii) |a| = 0 if and only if a = 0, (iii) |ab| = |a|·|b|, (iv) |a + b| ≤ max(|a|, |b|). The condition (iv) is called the strict triangle inequality. Because of (iii) the × × map | | : K −→ IR+ is a homomorphism of groups. In particular we have |1| = | − 1| = 1. We always will assume in addition that | | is non-trivial, i.e., that (v) there is an a0 ∈ K such that |a0| 6=0, 1. It follows immediately that |n · 1| ≤ 1 for any n ∈ ZZ. Moreover, if |a| 6= |b| for some a,b ∈ K then the strict triangle inequality actually can be sharpened into the equality |a + b| = max(|a|, |b|) . To see this we may assume that |a| < |b|. Then |a| < |b| = |b + a − a| ≤ max(|b+a|, |a|), hence |a| < |a+b| and therefore |b|≤|a+b| ≤ max(|a|, |b|) = |b|. Via the distance function d(a,b) := |b − a| the set K is a metric and hence topological space. The subsets Bǫ(a) := {b ∈ K : |b − a| ≤ ǫ} for any a ∈ K and any real number ǫ > 0 are called closed balls or simply balls in K. They form a fundamental system of neighbourhoods of a in the metric space K. Likewise the open balls − Bǫ (a) := {b ∈ K : |b − a| < ǫ} form a fundamental system of neighbourhoods of a in K. As we will see below − Bǫ(a) and Bǫ (a) are both open and closed subsets of K. Talking about open and closed balls therefore does not refer to a topological distinction but only to the nature of the inequality sign in the definition. We point out the following two simple facts. 1) If | |∞ denotes the usual archimedean absolute value on IR then, for any b ∈ − Bǫ (a), we have ||b|−|a||∞ = ||(b−a)+a|−|a||∞ ≤|max(|b−a|, |a|)−|a||∞ < ǫ. This means that the absolute value | | : K −→ IR is a continuous function. − − − 2) For b0 ∈ Bǫ (a0) and b1 ∈ Bǫ (a1) we have b0 + b1 ∈ Bǫ (a0 + a1) and b b ∈ B− (a a ). The latter follows from b b − a a = (b − 0 1 ǫ·max(ǫ,|a0|,|a1|) 0 1 0 1 0 1 0 a0)(b1 −a1)+(b0 −a0)a1 +a0(b1 −a1). This says that addition + : K ×K −→ K and multiplication · : K × K −→ K are continuous maps. 5 Lemma 1.1: i. Bǫ(a) is open and closed in K; ′ ′ ii. if Bǫ(a) ∩ Bǫ(a ) 6= ∅ then Bǫ(a) = Bǫ(a ); iii. If B and B′ are any two balls in K with B ∩ B′ 6= ∅ then either B ⊆ B′ or B′ ⊆ B; iv. K is totally disconnected. Proof: The assertions i. and ii. are immediate consequences of the strict triangle inequality. The assertion iii. follows from ii. To see iv. let M ⊆ K be a nonempty connected subset. Pick a point a ∈ M. By i. the intersection M ∩ Bǫ(a) is open and closed in M. It follows that M is contained in any ball around a and therefore must be equal to {a}. Clearly the assertions i.-iii. hold similarly for open balls. The assertion ii. says that any point of a (open) ball can serve as its midpoint. On the other hand the real number ǫ is not uniquely determined by the set Bǫ(a) and therefore cannot be considered as the ”radius” of this ball.
Load more

Recommended publications
  • Models of Linear Logic Based on the Schwartz $\Varepsilon $-Product
    MODELS OF LINEAR LOGIC BASED ON THE SCHWARTZ ε-PRODUCT. YOANN DABROWSKI AND MARIE KERJEAN Abstract. From the interpretation of Linear Logic multiplicative disjunction as the ε-product de- fined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. This improves on previous results in [BET] based on con- venient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality called k-reflexivity, allow- ing for a ∗-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the ε product. We adapt Meise’s definition of Smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same ∗-autonomous cat- egories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the ε-product (without reflexivization). Contents 1. Introduction 2 1.1. A first look at the interpretation of Linear Logic constructions 6 Part 1. Three Models of MALL 7 2. Preliminaries 7 2.1. Reminder on topological vector spaces 7 2.2. Reminder on tensor products and duals of locally convex spaces. 8 2.3. Dialogue and ∗-autonomous categories 10 2.4.
    [Show full text]
  • Marie Kerjean
    Tensor products and *-autonomous categories Marie Kerjean Laboratory PPS, Universit´eParis Diderot [email protected] The main use of ∗-autonomous categories is in the semantic study of Linear Logic. For this reason, it is thus natural to look for a ∗-autonomous category of locally convex topological vector spaces (tvs). On one hand, Linear Logic inherits its semantics from Linear Algebra, and it is thus natural to build models of Linear Logic from vector spaces [3,5,6,4]. On the other hand, denotational semantics has sought continuous models of computation through Scott domains [9]. Moreover, the infinite nature of the exponential of Linear Logic calls for infinite dimensional spaces, for which topology becomes necessary. One of the first intuitions that comes to mind when thinking about objects in a ∗-autonomous category is the notion of reflexive vector space, i.e. a a tvs which equals its double dual. When A is a vector space, the transpose dA : A ! (A !?) !? of the evaluation map evA :(A !?) × A !? is exactly the canonical injection of a vector space in its bidual. Then, requiring dA to be an isomorphism amounts to requiring A to be reflexive. However, the category of reflexive topological vector spaces is not ∗-autonomous, as it is not closed. Barr [2] constructs two closed subcategories of the category of tvs by re- stricting to tvs endowed with their weak topology (wtvs) or with their Mackey topology (mtvs), which are both polar topologies. Indeed, if E is a tvs, one can define its dual E0 as the space of all continuous linear form on E.
    [Show full text]
  • Weak Topologies for Linear Logic
    Weak topologies for Linear Logic Marie Kerjean Laboratoire PPS, Universit´eParis Diderot, Paris, France [email protected] Abstract We construct a denotational model of Linear Logic, whose objects are all the locally convex and separated topological vector spaces endowed with their weak topology. Linear proofs are interpreted as continuous linear functions, and non-linear proofs as sequences of monomials. The duality in this interpretation of Linear Logic does not come from an orthogonality relation, thus we do not complete our constructions by a double-orthogonality operation. This yields an interpretation of polarities with respect to weak topologies. 1 Introduction Linear Logic [Gir88] can be seen as a fine analysis of classical logic, through polarities and involutive linear negation [LR03]. The linearity hypothesis has been made by Girard [Gir87] after a semantical investigation of Intuitionistic Logic. Semantics has in turn led to various discoveries around Linear Logic, as in Game Semantics or Differential λ-calculus [ER06]. However, the linear negation is often modelized with an orthogonality relation [Ehr02, Ehr05, Gir04] or with a Chu construction [Gir99]. We try to generalize this approach by considering a model whose objects are general topological vector spaces. It allows us to get closer to the algebraic intuitions of Linear Logic, and to reach analogies with functional analysis. As in Scott Domains, we interpret our functions by continuous functions, and especially our linear proofs will be interpreted by linear continuous functions between topological vector spaces. As the topological dual of a space E is not constructed from E with an orthogonality relation, we have the opportunity to construct a new kind of negation.
    [Show full text]
  • NONARCHIMEDEAN COALGEBRAS and COADMISSIBLE MODULES 2 of Y
    NONARCHIMEDEAN COALGEBRAS AND COADMISSIBLE MODULES ANTON LYUBININ Abstract. We show that basic notions of locally analytic representation the- ory can be reformulated in the language of topological coalgebras (Hopf alge- bras) and comodules. We introduce the notion of admissible comodule and show that it corresponds to the notion of admissible representation in the case of compact p-adic group. Contents Introduction 1 1. Banach coalgebras 4 1.1. Banach -Coalgebras 5 ̂ 1.2. Constructions⊗ in the category of Banach -coalgebras 6 ̂ 1.3. Banach -bialgebras and Hopf -algebras⊗ 8 ̂ ̂ 1.4. Constructions⊗ in the category of⊗ Banach -bialgebras and Hopf ̂ -algebras. ⊗ 9 ̂ 2. Banach comodules⊗ 9 2.1. Basic definitions 9 2.2. Constructions in the category of Banach -comodules 10 ̂ 2.3. Induction ⊗ 11 2.4. Rational -modules 14 ̂ 2.5. Tensor identities⊗ 15 3. Locally convex -coalgebras 16 ̂ Preliminaries ⊗ 16 3.1. Topological Coalgebras 18 3.2. Topological Bialgebras and Hopf algebras. 20 4. modules and comodules 21 arXiv:1410.3731v2 [math.RA] 26 Jul 2017 4.1. Definitions 21 4.2. Rationality 22 4.3. Quotients, subobjects and simplicity 22 4.4. Cotensor product 23 5. Admissibility 24 Appendix 28 References 29 Introduction The study of p-adic locally analytic representation theory of p-adic groups seems to start in 1980s, with the first examples of such representations studied in the works 1 NONARCHIMEDEAN COALGEBRAS AND COADMISSIBLE MODULES 2 of Y. Morita [M1, M2, M3] (and A. Robert, around the same time), who considered locally analytic principal series representations for p-adic SL2.
    [Show full text]
  • Arxiv:1811.04430V2 [Math.OA]
    TRACES AND PEDERSEN IDEALS OF TENSOR PRODUCTS OF NONUNITAL C*-ALGEBRAS, CRISTIAN IVANESCU AND DAN KUCEROVSKˇ Y´ Abstract. We show that positive elements of a Pedersen ideal of a tensor product can be approximated in a particularly strong sense by sums of tensor products of positive elements. This has a range of applications to the structure of tracial cones and related topics, such as the Cuntz-Pedersen space or the Cuntz semigroup. For example, we determine the cone of lower semicontinuous traces of a tensor product in terms of the traces of the tensor factors, in an arbitrary C*-tensor norm. We show that the positive elements of a Pedersen ideal are sometimes stable under Cuntz equivalence. We generalize a result of Pedersen’s by showing that certain classes of completely positive maps take a Pedersen ideal into a Pedersen ideal. We provide theorems that in many cases compute the Cuntz semigroup of a tensor product. Key words and phrases. C∗-algebra, tensor product. 1. Introduction A unital C∗-algebra is a noncommutative generalization of the algebra C(X) of continuous functions on a compact topological space; in the nonunital case, it is a generalization of the algebra C0(X) of continuous functions that vanish at infinity, or more accurately, functions that are each arbitrarily small outside a sufficient large compact set. This paper studies the structure and applications of the Pedersen ideals of tensor products of nonunital C∗- algebras. The Pedersen ideal of a nonunital C∗-algebra is a noncommutative analogue of the space of compactly supported functions in the space of continuous complex-valued functions on a locally compact Hausdorff space.
    [Show full text]
  • Models of Linear Logic Based on the Schwartz Ε-Product
    Theory and Applications of Categories, Vol. 34, No. 45, 2019, pp. 1440–1525. MODELS OF LINEAR LOGIC BASED ON THE SCHWARTZ "-PRODUCT. YOANN DABROWSKI AND MARIE KERJEAN Abstract. From the interpretation of Linear Logic multiplicative disjunction as the epsilon product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on the usual mathematical notions of smooth maps. This improves on previous results by Blute, Ehrhard and Tasson based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated no- tion which is stable under duality called k-reflexivity, allowing for a star-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the epsilon- product. We adapt Meise’s definition of smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with con- veniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same star-autonomous cate- gories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the epsilon-product (without reflexivization). Contents 1 Introduction 1441 I. Models of MALL 1447 2 Preliminaries 1447 3 The original setting for the Schwartz "-product and smooth maps. 1457 4 Models of MALL coming from classes of smooth maps 1471 5 Schwartz locally convex spaces, Mackey-completeness and the ρ-dual.
    [Show full text]
  • Allanyashinski-Dissertation.Pdf
    The Pennsylvania State University The Graduate School PERIODIC CYCLIC HOMOLOGY AND SMOOTH DEFORMATIONS A Dissertation in Mathematics by Allan Yashinski c 2013 Allan Yashinski Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013 The dissertation of Allan Yashinski was reviewed and approved∗ by the following: Nigel Higson Evan Pugh Professor of Mathematics Dissertation Advisor, Chair of Committee Nathanial Brown Associate Professor of Mathematics John Roe Professor of Mathematics Radu Roiban Professor of Physics Yuxi Zheng Professor of Mathematics Department Head ∗Signatures are on file in the Graduate School. Abstract Given a formal deformation of an algebra, Getzler defined a connection on the periodic cyclic homology of the deformation, which he called the Gauss-Manin connection. We define and study this connection for smooth one-parameter defor- mations. Our main example is the smooth noncommutative n-torus AΘ, viewed as a deformation of the algebra C1(Tn) of smooth functions on the n-torus. In this case, we use the Gauss-Manin connection to give a parallel translation argument that shows that the periodic cyclic homology groups HP•(AΘ) are independent of the parameter Θ. As a consequence, we obtain differentiation formulas relating various cyclic cocycles on AΘ. We generalize this to a larger class of deformations, including nontrivial crossed product algebras by the group R. The algebras of such a deformation extend nat- urally to differential graded algebras, and we show that they are fiberwise isomor- phic as A1-algebras. In particular, periodic cyclic homology is preserved under this type of deformation. This clarifies and strengthens the periodic cyclic ho- mology isomorphism for noncommutative tori, and gives another proof of Connes' Thom isomorphism in cyclic homology.
    [Show full text]
  • The Space $\Dot {\Mathcal {B}}'$ of Distributions Vanishing at Infinity
    The space B˙′ of distributions vanishing at infinity – duals of tensor products E. A. Nigsch,∗ N. Ortner† April 11, 2016 Abstract Analogous to L. Schwartz’ study of the space D′(E) of semi-regular distributions we investigate the topological properties of the space D′(B˙) of semi-regular vanishing distributions and give representations of its dual and of the scalar product with this dual. In order to determine the dual of the space of semi-regular vanishing distributions we generalize and modify a result of A. Grothendieck on the duals of E⊗F if E and F are quasi-complete and F is not necessarily semi- reflexive. b Keywords: semi-regular vanishing distributions, duals of tensor products MSC2010 Classification: 46A32, 46F05 1 Introduction arXiv:1604.02846v1 [math.FA] 11 Apr 2016 L. Schwartz investigated, in his theory of vector-valued distributions [24, 25], ′ ′ Rn Rm ′ several subspaces of the space Dxy = D ( x × y ) that are of the type Dx(Ey) where Ey is a distribution space. Three prominent examples are: ∗Wolfgang-Pauli-Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. e-mail: [email protected]. Corresponding author. †Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. 1 ′ ′ (i) Dx(Sy) – the space of semi-temperate distributions, for which the par- tial Fourier transform is defined [24, p. 123]; ′ ′ (ii) Dx(DL1,y) – the space of partially summable distributions, for which the convolution of kernels is defined [7, §2, p. 546], as a generalization of the convolvability condition for two distributions [24, pp. 131,132]; ′ (iii) Dx(Ey) – the space of semi-regular distributions (see [23] and [24, p.
    [Show full text]
  • Nonstandard Analysis and Vector Lattices Managing Editor
    Nonstandard Analysis and Vector Lattices Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 525 Nonstandard Analysis and Vector Lattices Edited by S.S. Kutateladze Sobolev Institute of Mathematics. Siberian Division of the Russian Academy of Sciences. Novosibirsk. Russia SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C.LP. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-5863-6 ISBN 978-94-011-4305-9 (eBook) DOI 10.1007/978-94-011-4305-9 This is an updated translation of the original Russian work. Nonstandard Analysis and Vector Lattices, A.E. Gutman, \'E.Yu. Emelyanov, A.G. Kusraev and S.S. Kutateladze. Novosibirsk, Sobolev Institute Press, 1999. The book was typeset using AMS-TeX. Printed an acid-free paper AII Rights Reserved ©2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Foreword ix Chapter 1. Nonstandard Methods and Kantorovich Spaces (A. G. Kusraev and S. S. Kutateladze) 1 § 1.l. Zermelo-Fraenkel Set·Theory 5 § l.2. Boolean Valued Set Theory 7 § l.3. Internal and External Set Theories 12 § 1.4. Relative Internal Set Theory 18 § l.5. Kantorovich Spaces 23 § l.6. Reals Inside Boolean Valued Models 26 § l.7. Functional Calculus in Kantorovich Spaces 30 § l.8.
    [Show full text]
  • C~-Algebras from the Functional Analytic View Point
    Journal of Pure and Applied Algebra 46 (1987) 89-107 89 North-Holland C~-ALGEBRAS FROM THE FUNCTIONAL ANALYTIC VIEW POINT G. KAINZ, A. KRIEGL and P. MICHOR Institut fiir Mathematik, Universit?it Wien, Strudlhofgasse 4, A-1090 Wien, Austria Communicated by F.W. Lawvere Received June 1985 The notion of C~-algebra goes back to Lawvere [8], although the main examples appear much earlier in differential topology, singularity theory, and counter- examples can be found in harmonic analysis. Recently the notion of C~-algebra became important for the foundations of synthetic differential geometry: Kock [5], Dubuc [1], and the forthcoming book of Moerdijk and Reyes [10]. In this paper we put a 'natural' locally-m-convex topology on each C~-algebra, using the framework of convenient vector spaces of Kriegl [6, 7]. For important examples this topology is non-Hausdorff (germs of flat functions are always cluster points of 0), but if it is Hausdorff, we are able to derive nice results: The C~-homomorphisms are exactly the continuous algebra homomorphisms, countably generated Hausdorff C~-algebras are nuclear, the action of smooth functions on Hausdorff C~-algebras is smooth and continuous, and the coproduct equals the bornological tensor product in the most important cases. We also con- sider C~-modules. Working on this paper we came to believe that the notion of C~-algebra is ade- quate as long as it is finitely generated. Infinitely generated algebras should at least be product preserving functors on convenient vector spaces, say. Then the free ones would be the 'usual' spaces of smooth functions on products of factors ~.
    [Show full text]
  • An Adjunction Formula for the Emerton-Jacquet Functor John Bergdall Bryn Mawr College, [email protected]
    Bryn Mawr College Scholarship, Research, and Creative Work at Bryn Mawr College Mathematics Faculty Research and Scholarship Mathematics 2018 An adjunction formula for the Emerton-Jacquet functor John Bergdall Bryn Mawr College, [email protected] Przemyslaw Chojecki Let us know how access to this document benefits ouy . Follow this and additional works at: https://repository.brynmawr.edu/math_pubs Part of the Mathematics Commons Custom Citation Berdgall, John and Przemyslaw Chojecki. 2018. "An adjunction formula for the Emerton-Jacquet functor." Israel Journal of Mathematics, 223.1: 1-52, 2018. This paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. https://repository.brynmawr.edu/math_pubs/18 For more information, please contact [email protected]. AN ADJUNCTION FORMULA FOR THE EMERTON{JACQUET FUNCTOR JOHN BERGDALL AND PRZEMYSLAW CHOJECKI Abstract. The Emerton{Jacquet functor is a tool for studying locally analytic represen- tations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton{Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting. Contents 1. Introduction 1 2. Reminders 6 3. Composition series of some Verma modules 11 4. The adjunction formula 24 5. Restriction to the socle 32 References 35 1. Introduction Let p be a prime. Throughout this paper we fix a finite extension K=Qp and let G be a connected, reductive and split algebraic group defined over K.
    [Show full text]
  • [Math.KT] 29 Jun 1999 Nltccci Cohomology Cyclic Analytic Afmeyer Ralf 1999 Contents
    Ralf Meyer Analytic cyclic cohomology 1999 arXiv:math/9906205v1 [math.KT] 29 Jun 1999 Contents 1 Introduction 4 2 Bornologies 8 2.1 BasicDefinitions ................................. .... 9 2.1.1 Bornologies ................................... 9 2.1.2 Boundedmaps .................................. 10 2.1.3 Bornologicalconvergence . ..... 11 2.2 Constructions with bornological vector spaces . .............. 12 2.2.1 Subspaces,quotients,extensions . ........ 12 2.2.2 Completions ................................... 13 2.2.3 Completed bornologicaltensor products . ......... 14 2.2.4 Spacesofboundedlinearmaps . .... 16 2.2.5 Smooth and absolutely continuous homotopies . ......... 16 3 Analytic cyclic cohomology 18 3.1 Analytic tensor algebras and a-nilpotent algebras . ............... 18 3.1.1 Definition of the analytic tensor algebra . ......... 21 3.1.2 Properties of analytically nilpotent algebras . ............. 23 3.1.3 The interrelations between analytic nilpotence, lanilcurs, and analytic tensor algebras...................................... 25 3.1.4 Reformulations of the Extension and the Homotopy Axiom ......... 28 3.1.5 Universal analytically nilpotent extensions . ............. 30 3.1.6 The bimodule Ω1(T A).............................. 32 3.1.7 The lanilcur category and Goodwillie’s theorem . ........... 34 3.2 The X-complex of T A andanalyticcycliccohomology . 36 3.2.1 TheX-complexofaquasi-freealgebra . ....... 36 3.2.2 Definition and functoriality of analytic cyclic cohomology .......... 37 3.2.3 Homotopy invariance and stability . ....... 38 3.2.4 Adjoiningunits................................ .. 40 3.2.5 The Chern-Connes character in K-theory ................... 42 3.2.6 The X-complex of T A andentirecycliccohomology . 44 3.3 Excisioninanalyticcycliccohomology . ........... 46 3.3.1 Outlineoftheproof ............................. .. 47 3.3.2 Linear functoriality of Ωan ............................ 48 3.3.3 Someisomorphisms .............................. 49 3.3.4 A free resolution of L+ .............................
    [Show full text]