the dual space of precompact groups
play

The dual space of precompact groups Salvador Hern andez - PowerPoint PPT Presentation

Sep 29, 2023 •131 likes •840 views

The dual space of precompact groups Salvador Hern andez Universitat Jaume I The dual space of precompact groups - Presented at the AHA 2013 Conference Granada, May 20 - 24, 2013. - Joint work with M. Ferrer and V. Uspenskij . . . . .

  1. The dual space of precompact groups Salvador Hern´ andez Universitat Jaume I The dual space of precompact groups - Presented at the AHA 2013 Conference Granada, May 20 - 24, 2013. - Joint work with M. Ferrer and V. Uspenskij . . . . . .

  2. Index 1 Introduction . . . . . .

  3. Index 1 Introduction 2 Notation and basic facts . . . . . .

  4. Index 1 Introduction 2 Notation and basic facts 3 Precompact metrizable groups . . . . . .

  5. Index 1 Introduction 2 Notation and basic facts 3 Precompact metrizable groups 4 Discrete metrics . . . . . .

  6. Index 1 Introduction 2 Notation and basic facts 3 Precompact metrizable groups 4 Discrete metrics 5 Non-metrizable precompact groups . . . . . .

  7. Index 1 Introduction 2 Notation and basic facts 3 Precompact metrizable groups 4 Discrete metrics 5 Non-metrizable precompact groups 6 Property (T) . . . . . .

  8. Index 1 Introduction 2 Notation and basic facts 3 Precompact metrizable groups 4 Discrete metrics 5 Non-metrizable precompact groups 6 Property (T) . . . . . .

  9. Introduction In this talk we are concerned with the extension to topological groups of following classical result. Theorem (Banach - Dieudonn´ e) If E is a metrizable locally convex space, the precompact-open topology on its dual E ′ coincides with the topology of N -convergence, where N is the collection of all compact subsets of E each of which is the set of points of a sequence converging to 0. . . . . . .

  10. Introduction In this talk we are concerned with the extension to topological groups of following classical result. Theorem (Banach - Dieudonn´ e) If E is a metrizable locally convex space, the precompact-open topology on its dual E ′ coincides with the topology of N -convergence, where N is the collection of all compact subsets of E each of which is the set of points of a sequence converging to 0. So far, this result had been extended to metrizable abelian groups by several authors: Banaszczyk (1991) for metrizable vector groups, Aussenhofer (1999) and, independently, Chasco (1998) for metrizable abelian groups. . . . . . .

  11. Introduction In this talk we are concerned with the extension to topological groups of following classical result. Theorem (Banach - Dieudonn´ e) If E is a metrizable locally convex space, the precompact-open topology on its dual E ′ coincides with the topology of N -convergence, where N is the collection of all compact subsets of E each of which is the set of points of a sequence converging to 0. So far, this result had been extended to metrizable abelian groups by several authors: Banaszczyk (1991) for metrizable vector groups, Aussenhofer (1999) and, independently, Chasco (1998) for metrizable abelian groups. I’m going to report on our findings concerning the extension of the Banach - Dieudonn´ e Theorem to non necessarily abelian, metrizable, precompact groups. . . . . . .

  12. Index 1 Introduction 2 Notation and basic facts 3 Precompact metrizable groups 4 Discrete metrics 5 Non-metrizable precompact groups 6 Property (T) . . . . . .

  13. Notation and basic facts For a topological group G , let � G be the set of equivalence classes of irreducible unitary representations of G . The set � G can be equipped with a natural topology, the so-called Fell topology. . . . . . .

  14. Notation and basic facts For a topological group G , let � G be the set of equivalence classes of irreducible unitary representations of G . The set � G can be equipped with a natural topology, the so-called Fell topology. If G is Abelian, then � G is the standard Pontryagin-van Kampen dual group and the Fell topology on � G is the usual compact-open topology; . . . . . .

  15. Notation and basic facts For a topological group G , let � G be the set of equivalence classes of irreducible unitary representations of G . The set � G can be equipped with a natural topology, the so-called Fell topology. If G is Abelian, then � G is the standard Pontryagin-van Kampen dual group and the Fell topology on � G is the usual compact-open topology; When G is compact, the Fell topology on � G is the discrete topology; . . . . . .

  16. Notation and basic facts For a topological group G , let � G be the set of equivalence classes of irreducible unitary representations of G . The set � G can be equipped with a natural topology, the so-called Fell topology. If G is Abelian, then � G is the standard Pontryagin-van Kampen dual group and the Fell topology on � G is the usual compact-open topology; When G is compact, the Fell topology on � G is the discrete topology; When � G is neither Abelian nor compact, � G usually is non-Hausdorff. . . . . . .

  17. Notation and basic facts For a topological group G , let � G be the set of equivalence classes of irreducible unitary representations of G . The set � G can be equipped with a natural topology, the so-called Fell topology. If G is Abelian, then � G is the standard Pontryagin-van Kampen dual group and the Fell topology on � G is the usual compact-open topology; When G is compact, the Fell topology on � G is the discrete topology; When � G is neither Abelian nor compact, � G usually is non-Hausdorff. In general, little is known about the properties of the Fell topology. . . . . . .

  18. Notation and basic facts A topological group G is precompact if it is isomorphic (as a topological group) to a subgroup of a compact group H (we may assume that G is dense in H ). . . . . . .

  19. Notation and basic facts A topological group G is precompact if it is isomorphic (as a topological group) to a subgroup of a compact group H (we may assume that G is dense in H ). If G is a dense subgroup of a compact group H , the precompact-open topology on � G coincides with the compact-open topology on � H . . . . . . .

  20. Notation and basic facts A topological group G is precompact if it is isomorphic (as a topological group) to a subgroup of a compact group H (we may assume that G is dense in H ). If G is a dense subgroup of a compact group H , the precompact-open topology on � G coincides with the compact-open topology on � H . Since the dual space of a compact group is discrete, in order to prove that a precompact group G satisfies the Banach - Dieudonn´ e Theorem, it suffices to verify that � G is discrete. . . . . . .

  21. Notation and basic facts A topological group G is precompact if it is isomorphic (as a topological group) to a subgroup of a compact group H (we may assume that G is dense in H ). If G is a dense subgroup of a compact group H , the precompact-open topology on � G coincides with the compact-open topology on � H . Since the dual space of a compact group is discrete, in order to prove that a precompact group G satisfies the Banach - Dieudonn´ e Theorem, it suffices to verify that � G is discrete. Thus, we look at the following question: for what precompact groups G is � G discrete? . . . . . .

  22. Dual object Two unitary representations ρ : G → U ( H 1 ) and ψ : G → U ( H 2 ) are equivalent if there exists a Hilbert space isomorphism M : H 1 → H 2 such that ρ ( x ) = M − 1 ψ ( x ) M for all x ∈ G . . . . . . .

  23. Dual object Two unitary representations ρ : G → U ( H 1 ) and ψ : G → U ( H 2 ) are equivalent if there exists a Hilbert space isomorphism M : H 1 → H 2 such that ρ ( x ) = M − 1 ψ ( x ) M for all x ∈ G . The dual object of G is the set � G of equivalence classes of irreducible unitary representations of G . . . . . . .

  24. Dual object Two unitary representations ρ : G → U ( H 1 ) and ψ : G → U ( H 2 ) are equivalent if there exists a Hilbert space isomorphism M : H 1 → H 2 such that ρ ( x ) = M − 1 ψ ( x ) M for all x ∈ G . The dual object of G is the set � G of equivalence classes of irreducible unitary representations of G . If G is a compact group, all irreducible unitary representation of G are finite-dimensional and the Peter-Weyl Theorem determines an embedding of G into the product of unitary groups U ( n ). . . . . . .

  25. Functions of positive type If ρ : G → U ( H ) is a unitary representation, a complex-valued function f on G is called a function of positive type associated with ρ if there exists a vector v ∈ H such that f ( g ) = ( ρ ( g ) v , v ) ∀ g ∈ G . . . . . .

  26. Functions of positive type If ρ : G → U ( H ) is a unitary representation, a complex-valued function f on G is called a function of positive type associated with ρ if there exists a vector v ∈ H such that f ( g ) = ( ρ ( g ) v , v ) ∀ g ∈ G We denote by P ′ ρ be the set of all functions of positive type associated with ρ . Let P ρ be the convex cone generated by P ′ ρ . . . . . . .

  27. Functions of positive type If ρ : G → U ( H ) is a unitary representation, a complex-valued function f on G is called a function of positive type associated with ρ if there exists a vector v ∈ H such that f ( g ) = ( ρ ( g ) v , v ) ∀ g ∈ G We denote by P ′ ρ be the set of all functions of positive type associated with ρ . Let P ρ be the convex cone generated by P ′ ρ . If ρ 1 and ρ 2 are equivalent representations, then P ′ ρ 1 = P ′ ρ 2 and P ρ 1 = P ρ 2 . . . . . . .

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

The Baire property Introduction A characterization of on precompact abelian groups the Baire

The Baire property Introduction A characterization of on precompact abelian groups the Baire

The Baire property on precompact abelian groups The Baire property Introduction A characterization of on precompact abelian groups the Baire property Compact subsets of a precompact, bounded Baire group X. Domnguez Baire subgroups of T

396 views • 35 slides
Property (T) for quantum groups from the Property (T) for groups dual point of view Quantum

Property (T) for quantum groups from the Property (T) for groups dual point of view Quantum

Property (T) for quantum groups from the dual point of view David Kyed Property (T) for quantum groups from the Property (T) for groups dual point of view Quantum groups Property (T) for quantum groups David Kyed The dual picture

936 views • 61 slides
A second order finite volume scheme on space-adaptive staggered grids in 3D Wolfram Rosenbaum

A second order finite volume scheme on space-adaptive staggered grids in 3D Wolfram Rosenbaum

1 A second order finite volume scheme on space-adaptive staggered grids in 3D Wolfram Rosenbaum Prof. Dr. Sebastian Noelle RWTH Aachen, Germany Staggered grid construction in 1D 2 How to construct dual cells dual cell dual grid primal

510 views • 21 slides
On Self-dual Codes Darwin Villar July 2018 Introduction Codes and monomial groups Generalized

On Self-dual Codes Darwin Villar July 2018 Introduction Codes and monomial groups Generalized

On Self-dual Codes Darwin Villar July 2018 Introduction Codes and monomial groups Generalized Code [52,26,15] Generalization Open Questions Introduction Let C be a self-dual [ n , k , d ]- code over F q . Type I C is 2 -divisible or even

249 views • 6 slides
Fundamental groups of complements of dual varieties in Grass- mannian Hakata, 2007 September

Fundamental groups of complements of dual varieties in Grass- mannian Hakata, 2007 September

Fundamental groups of complements of dual varieties in Grass- mannian Hakata, 2007 September Ichiro Shimada (Hokkaido University, Sapporo, JAPAN) 1 1. Introduction This work is motivated by the conjecture in the paper [ADKY] D. Auroux, S.

560 views • 32 slides
Fundamental groups of complements of dual varieties in Grassmannian Kyoto, RIMS, 2007 July

Fundamental groups of complements of dual varieties in Grassmannian Kyoto, RIMS, 2007 July

Fundamental groups of complements of dual varieties in Grassmannian Kyoto, RIMS, 2007 July Ichiro Shimada (Hokkaido University, Sapporo, JAPAN) 1 1. Introduction This work is motivated by the conjecture in the paper [ADKY] D. Auroux, S.

579 views • 24 slides
1 Usages for From-Region visibility Conservative Visibility Sets Amortization: Exact

1 Usages for From-Region visibility Conservative Visibility Sets Amortization: Exact

From-Region Visibility and Overview Ray Space Factorization Short introduction to the problem Daniel Cohen-Or Dual Space & Parameter/Ray Space Tel-Aviv University Ray space factorization (SIGGRAPH03) From Point Visibility

546 views • 13 slides
Dual System Encryption Framework in Prime-Order Groups via Computational Pair Encodings

Dual System Encryption Framework in Prime-Order Groups via Computational Pair Encodings

1 Dual System Encryption Framework in Prime-Order Groups via Computational Pair Encodings Nuttapong Attrapadung (Nuts) AIST, Japan Asiacrypt 2016 Hanoi, Vietnam, December 7, 2016 2 Our Main Result in One Slide A Generic Framework for

634 views • 60 slides
Coxeter groups and Artin groups Day 2: Symmetric Spaces and Simple Groups Jon McCammond (U.C.

Coxeter groups and Artin groups Day 2: Symmetric Spaces and Simple Groups Jon McCammond (U.C.

Coxeter groups and Artin groups Day 2: Symmetric Spaces and Simple Groups Jon McCammond (U.C. Santa Barbara) 1 Symmetric Spaces Def: A Riemannian manifold is called a symmetric space if at each point p there exists an isometry fixing p and

295 views • 25 slides
Constant mean curvature surfaces in Minkowski 3-space via loop groups David Brander Now:

Constant mean curvature surfaces in Minkowski 3-space via loop groups David Brander Now:

CMC Surfaces in Euclidean Space CMC surfaces in Minkowski 3-Space Constant mean curvature surfaces in Minkowski 3-space via loop groups David Brander Now: Department of Mathematics Kobe University (From August 2008: Danish Technical

856 views • 63 slides
Handouts on CNSC website 3 1 How to submit your Feedback about todays session 4 Survey Monkey

Handouts on CNSC website 3 1 How to submit your Feedback about todays session 4 Survey Monkey

Concurrent Disorder Groups for Individuals with Dual Diagnosis It Can Be Done! Melvin Ong, BSSc, Dual Diagnosis Justice Case Manager mong@cmhaottawa.ca Lisa St. Pierre, BA (Hons), MSW Program Manager lstpierre@cmhaottawa.ca Craig Defries, MSW,

473 views • 15 slides
Witt groups and Witt spaces 1. Geometric interpretations of the Witt groups W ( Z ) and W ( Q )

Witt groups and Witt spaces 1. Geometric interpretations of the Witt groups W ( Z ) and W ( Q )

Witt groups and Witt spaces 1. Geometric interpretations of the Witt groups W ( Z ) and W ( Q ) 2. Witt space bordism and BalmerWitt groups of PL-constructible sheaves 3. Witt bordism proof of CappellShanesons L-class formula 4.

201 views • 17 slides
Minimal homeomorphisms of a Cantor space: full groups and invariant measures J. Melleray

Minimal homeomorphisms of a Cantor space: full groups and invariant measures J. Melleray

Minimal homeomorphisms of a Cantor space: full groups and invariant measures J. Melleray Institut Camille Jordan (Universit e de Lyon) Toposym 2016, Prague J. Melleray Full groups of minimal homeomorphisms Joint work with Tom as

859 views • 64 slides
Ever experienced problems with congestion pricing using LMPs (byproduct of optimal dispatch)?

Ever experienced problems with congestion pricing using LMPs (byproduct of optimal dispatch)?

Ever experienced problems with congestion pricing using LMPs (byproduct of optimal dispatch)? Dual space shows the other side of the coin! $ MW A dual space approach for pricing congestion in electricity markets Ricardo

400 views • 37 slides
Dual Credit Courses What does it mean to be a dual credit student? Dual enrollment simply means: A

Dual Credit Courses What does it mean to be a dual credit student? Dual enrollment simply means: A

Lamar Consolidated ISD & Lone Star College Present Dual Credit Courses What does it mean to be a dual credit student? Dual enrollment simply means: A student is still in high school and enrolled in a college-level class at the same time;

371 views • 16 slides
DUAL CREDIT WHAT IS DUAL CREDIT? Dual credit means two things are happening at once. Students

DUAL CREDIT WHAT IS DUAL CREDIT? Dual credit means two things are happening at once. Students

DUAL CREDIT WHAT IS DUAL CREDIT? Dual credit means two things are happening at once. Students are earning high school credit while also earning college credit. Dual credit courses are taught at the college level. Expect the class to be

323 views • 21 slides
Van Kampen diagrams are bicolimits Pawel Sobocinski, University of Southampton (joint work with

Van Kampen diagrams are bicolimits Pawel Sobocinski, University of Southampton (joint work with

Van Kampen diagrams are bicolimits Pawel Sobocinski, University of Southampton (joint work with Tobias Heindel) CT 08, Calais, 23/06/08 and Span( C ) C (Benabou) in this talk category = category with pullbacks : C Span( C ) f f C

624 views • 17 slides
Homotopy groups, braids and links Jie Wu National University of Singapore Novosibirsk workshop

Homotopy groups, braids and links Jie Wu National University of Singapore Novosibirsk workshop

Homotopy Groups Combinatorial Aspects of Homotopy Groups Homotopy and Braids Homotopy and Links Homotopy groups, braids and links Jie Wu National University of Singapore Novosibirsk workshop on "Knots, Braids, and Automorphism

2.08k views • 23 slides
Cotorsion-free groups from a topological viewpoint Hanspeter Fischer (Ball State University, USA)

Cotorsion-free groups from a topological viewpoint Hanspeter Fischer (Ball State University, USA)

Introduction Results Proofs Cotorsion-free groups from a topological viewpoint Hanspeter Fischer (Ball State University, USA) joint work with Katsuya Eda (Waseda University, Japan) TOPOSYM 2016 Prague, Czech Republic July 26, 2016 K. Eda,

1.16k views • 95 slides
Automorphisms of relatively hyperbolic groups Gilbert Levitt (joint work with V. Guirardel, A.

Automorphisms of relatively hyperbolic groups Gilbert Levitt (joint work with V. Guirardel, A.

Automorphisms of relatively hyperbolic groups Gilbert Levitt (joint work with V. Guirardel, A. Minasyan) Universit e de Caen Paris, juillet 2011 Gilbert Levitt Automorphisms of relatively hyperbolic groups What to remember from this talk A

381 views • 11 slides
Bridge numbers of links and minimal numbers of meridian generators of link groups Yeonhee JANG

Bridge numbers of links and minimal numbers of meridian generators of link groups Yeonhee JANG

Bridge numbers of links and minimal numbers of meridian generators of link groups Yeonhee JANG Hiroshima University, JSPS research fellow CONTENTS 1. Introduction. 2. Result. 3. Outline of Proof. 4. Open Problems.

352 views • 11 slides
SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T.

SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T.

Introduction Free energy Thermodynamic integration Adaptive biasing techniques SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T. Lelivre CERMICS - Ecole des Ponts ParisTech & Matherials

847 views • 67 slides
The Phoibos Analyzer Series Electron Spectrometers for Hard x-ray Photoemission Spectroscopy

The Phoibos Analyzer Series Electron Spectrometers for Hard x-ray Photoemission Spectroscopy

Hamburg, 16. September 2011 Thorsten U. Kampen The Phoibos Analyzer Series Electron Spectrometers for Hard x-ray Photoemission Spectroscopy PHOIBOS analyzer series PHOIBOS 100 PHOIBOS 150 PHOIBOS 225 PHOIBOS 150 WAL PHOIBOS 150 NAP Power

428 views • 26 slides
Higher Structures in Algebraic Quantum Field Theory Alexander Schenkel School of Mathematical

Higher Structures in Algebraic Quantum Field Theory Alexander Schenkel School of Mathematical

Higher Structures in Algebraic Quantum Field Theory Alexander Schenkel School of Mathematical Sciences, University of Nottingham Mathematics of Interacting QFT Models, 15 July 2019, York. Based on joint works with M. Benini and different

1.36k views • 105 slides

More recommend