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Artykuły w czasopismach na temat „Dualité de Pontryagin”

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Data publikacji: 14 grudnia 2024

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1

Lim, Johnny. "Analytic Pontryagin duality." Journal of Geometry and Physics 145 (November 2019): 103483. http://dx.doi.org/10.1016/j.geomphys.2019.103483.

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2

Chasco, M. J., and E. Mart�n-Peinador. "Binz-Butzmann duality versus Pontryagin duality." Archiv der Mathematik 63, no. 3 (1994): 264–70. http://dx.doi.org/10.1007/bf01189829.

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3

Banaszczyk, Wojciech, María Jesús Chasco, and Elena Martin-Peinador. "Open subgroups and Pontryagin duality." Mathematische Zeitschrift 215, no. 1 (1994): 195–204. http://dx.doi.org/10.1007/bf02571709.

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4

Chasco, M. J. "Pontryagin duality for metrizable groups." Archiv der Mathematik 70, no. 1 (1998): 22–28. http://dx.doi.org/10.1007/s000130050160.

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5

Shtern, A. I. "Duality between compactness and discreteness beyond pontryagin duality." Proceedings of the Steklov Institute of Mathematics 271, no. 1 (2010): 212–27. http://dx.doi.org/10.1134/s0081543810040164.

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6

Melnikov, Alexander. "Computable topological groups and Pontryagin duality." Transactions of the American Mathematical Society 370, no. 12 (2018): 8709–37. http://dx.doi.org/10.1090/tran/7355.

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7

Hern�ndez, Salvador. "Pontryagin duality for topological Abelian groups." Mathematische Zeitschrift 238, no. 3 (2001): 493–503. http://dx.doi.org/10.1007/s002090100263.

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8

Van Daele, A., and Shuanhong Wang. "Pontryagin duality for bornological quantum hypergroups." manuscripta mathematica 131, no. 1-2 (2009): 247–63. http://dx.doi.org/10.1007/s00229-009-0318-8.

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9

Hernández, Salvador, and Vladimir Uspenskij. "Pontryagin Duality for Spaces of Continuous Functions." Journal of Mathematical Analysis and Applications 242, no. 2 (2000): 135–44. http://dx.doi.org/10.1006/jmaa.1999.6627.

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10

Gabriyelyan, S. S. "Groups of quasi-invariance and the Pontryagin duality." Topology and its Applications 157, no. 18 (2010): 2786–802. http://dx.doi.org/10.1016/j.topol.2010.08.018.

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11

Lai, King Fai, Ignazio Longhi, Ki-Seng Tan, and Fabien Trihan. "Pontryagin duality for Iwasawa modules and abelian varieties." Transactions of the American Mathematical Society 370, no. 3 (2017): 1925–58. http://dx.doi.org/10.1090/tran/7016.

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12

Akbarov, S. S. "Pontryagin duality in the theory of topological modules." Functional Analysis and Its Applications 29, no. 4 (1996): 276–79. http://dx.doi.org/10.1007/bf01077475.

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13

Sharma, Pranav. "Duality of locally quasi-convex convergence groups." Applied General Topology 22, no. 1 (2021): 193. http://dx.doi.org/10.4995/agt.2021.14585.

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<p>In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive. Further, we prove that every character group of a convergence group is locally quasi-convex.</p>
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14

Artusa, Marco. "Duality for condensed cohomology of the Weil group of a $p$-adic field." Documenta Mathematica 29, no. 6 (2024): 1381–434. http://dx.doi.org/10.4171/dm/977.

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We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a p -adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups.
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15

Glöckner, Helge, Ralf Gramlich, and Tobias Hartnick. "Final group topologies, Kac-Moody groups and Pontryagin duality." Israel Journal of Mathematics 177, no. 1 (2010): 49–101. http://dx.doi.org/10.1007/s11856-010-0038-5.

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16

Banaszczyk, Wojciech. "Pontryagin duality for subgroups and quotients of nuclear spaces." Mathematische Annalen 273, no. 4 (1986): 653–64. http://dx.doi.org/10.1007/bf01472136.

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17

Akbarov, S. S. "Pontryagin duality in the theory of topological vector spaces." Mathematical Notes 57, no. 3 (1995): 319–22. http://dx.doi.org/10.1007/bf02303980.

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18

Balmer, Paul, Ivo Dell’Ambrogio, and Beren Sanders. "Grothendieck–Neeman duality and the Wirthmüller isomorphism." Compositio Mathematica 152, no. 8 (2016): 1740–76. http://dx.doi.org/10.1112/s0010437x16007375.

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We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.
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19

Górka, Przemysław, and Tomasz Kostrzewa. "Pego everywhere." Journal of Algebra and Its Applications 15, no. 04 (2016): 1650074. http://dx.doi.org/10.1142/s0219498816500742.

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In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.
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20

Remus, Dieter, and F. Javier Trigos-Arrieta. "Abelian groups which satisfy Pontryagin duality need not respect compactness." Proceedings of the American Mathematical Society 117, no. 4 (1993): 1195. http://dx.doi.org/10.1090/s0002-9939-1993-1132422-4.

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21

Pestov, Vladimir. "Free Abelian topological groups and the Pontryagin-Van Kampen duality." Bulletin of the Australian Mathematical Society 52, no. 2 (1995): 297–311. http://dx.doi.org/10.1017/s0004972700014726.

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We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.
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22

Ardanza-Trevijano, S., and M. J. Chasco. "The Pontryagin duality of sequential limits of topological Abelian groups." Journal of Pure and Applied Algebra 202, no. 1-3 (2005): 11–21. http://dx.doi.org/10.1016/j.jpaa.2005.02.006.

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23

Becker, Christian, Marco Benini, Alexander Schenkel, and Richard J. Szabo. "Cheeger–Simons differential characters with compact support and Pontryagin duality." Communications in Analysis and Geometry 27, no. 7 (2019): 1473–522. http://dx.doi.org/10.4310/cag.2019.v27.n7.a2.

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24

Hollevoet, Jan. "Pontryagin Duality for a Class of Locally Compact Quantum Groups." Mathematische Nachrichten 176, no. 1 (1995): 93–110. http://dx.doi.org/10.1002/mana.19951760108.

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25

Quackenbush, R., and C. S. Szabó. "Nilpotent groups are not dualizable." Journal of the Australian Mathematical Society 72, no. 2 (2002): 173–80. http://dx.doi.org/10.1017/s1446788700003827.

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AbstractIt is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.
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26

CABRAL, L. A. "GEOMETRIC DUALITY AND CHERN–SIMONS MODIFIED GRAVITY." International Journal of Modern Physics D 19, no. 08n10 (2010): 1323–27. http://dx.doi.org/10.1142/s0218271810017664.

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We consider a theory which involves an extension of general relativity known as Chern–Simons modified gravity (CSMG). In this theory the standard Einstein–Hilbert action is extended with a gravitational Pontryagin density that is obtained from a divergence of a Chern–Simons topological current. The extended theory has the standard Schwarzchild metric as solution, however, only a perturbed Kerr metric holds solution. From the exact Kerr metric we construct dual metrics to search for rotating black hole solutions. The conditions on the Killing tensors associated with dual metrics entail nontrivial solutions to CSMG.
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27

REŞIT DÜNDARER, A. "OCTONIONIC MULTI S8 CHIRAL AND GRAVITATIONAL INSTANTONS." Modern Physics Letters A 06, no. 17 (1991): 1611–14. http://dx.doi.org/10.1142/s0217732391001743.

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An 8-dimensional generalization of the sigma model is given and it is shown that these fields have topological charge n and satisfy a self-duality equation for the octonionic mappings xn: S8 → S8. Furthermore the Euler–Poincaré index I E and the Pontryagin index I P are generalized to eight dimensions and it is shown that I E = n, I P = 0 for the above mappings.
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28

Thuillier, F. "3D topological models and Heegaard splitting. II. Pontryagin duality and observables." Journal of Mathematical Physics 61, no. 11 (2020): 112302. http://dx.doi.org/10.1063/5.0027779.

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29

Garibay Bonales, Fernando, F. Javier Trigos-Arrieta, and Rigoberto Vera Mendoza. "A characterization of Pontryagin–van Kampen duality for locally convex spaces." Topology and its Applications 121, no. 1-2 (2002): 75–89. http://dx.doi.org/10.1016/s0166-8641(01)00111-0.

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30

GEYER, BODO. "COHOMOLOGICAL GAUGE THEORIES IN D > 4 WITH SPECIAL HOLONOMY Spin(7) AND G2." International Journal of Modern Physics A 20, no. 11 (2005): 2490–99. http://dx.doi.org/10.1142/s0217751x05024821.

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From Minkowskian 10D super Yang-Mills theory, by dimensional reduction and continuous Weyl-rotation, the 8D Euclidean, cohomological Spin(7)-invariant action [Formula: see text] is derived, by reduction to 7D the cohomological G2-invariant action [Formula: see text] with global SU(2) symmetry obtains. Compatibility of chirality with generalized self-duality and octonionic algebra is shown. Using the chiral primary operator and the 8D analogue of the Pontryagin invariant a cohomological extension of [Formula: see text] has been constructed.
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31

Garibay Bonales, Fernando, F. Javier Trigos-Arrieta, and Rigoberto Vera Mendoza. "A CHARACTERIZATION OF PONTRYAGIN-VAN KAMPEN DUALITY FOR COMPLEX LOCALLY CONVEX SPACES." Communications in Algebra 30, no. 4 (2002): 1715–24. http://dx.doi.org/10.1081/agb-120013211.

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32

Bruguera, M., and M. Tkachenko. "Pontryagin duality in the class of precompact Abelian groups and the Baire property." Journal of Pure and Applied Algebra 216, no. 12 (2012): 2636–47. http://dx.doi.org/10.1016/j.jpaa.2012.03.035.

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33

Akbarov, S. S. "Kernels and cokernels in the category of augmented involutive stereotype algebras." Journal of Algebra and Its Applications 19, no. 06 (2019): 2050108. http://dx.doi.org/10.1142/s021949882050108x.

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We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: (1) this category has kernels and cokernels, (2) the cokernel is preserved under the passage to the group stereotype algebras, and (3) the notion of cokernel allows to prove that the continuous envelope [Formula: see text] of the group algebra of a compact buildup of an abelian locally compact group is an involutive Hopf algebra in the category of stereotype spaces [Formula: see text]. The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.
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34

Tamminen, Eero V. "Strong Lagrange duality and the maximum principle for nonlinear discrete time optimal control problems." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 20. http://dx.doi.org/10.1051/cocv/2018012.

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We examine discrete-time optimal control problems with general, possibly non-linear or non-smooth dynamic equations, and state-control inequality and equality constraints. A new generalized convexity condition for the dynamics and constraints is defined, and it is proved that this property, together with a constraint qualification constitute sufficient conditions for the strong Lagrange duality result and saddle-point optimality conditions for the problem. The discrete maximum principle of Pontryagin is obtained in a straightforward manner from the strong Lagrange duality theorem, first in a new form in which the Lagrangian is minimized both with respect to the state and to the control variables. Assuming differentiability, the maximum principle is obtained in the usual form. It is shown that dynamic systems satisfying a global controllability condition with convex costs, have the required convexity property. This controllability condition is a natural extension of the customary directional convexity condition applied in the derivation of the discrete maximum principle for local optima in the literature.
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35

Aristov, O. Yu. "On holomorphic reflexivity conditions for complex Lie groups." Proceedings of the Edinburgh Mathematical Society 64, no. 4 (2021): 800–821. http://dx.doi.org/10.1017/s0013091521000572.

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AbstractWe consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$.
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36

Dikranjan, Dikran, and Luchezar Stoyanov. "An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups." Topology and its Applications 158, no. 15 (2011): 1942–61. http://dx.doi.org/10.1016/j.topol.2011.06.037.

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37

Chis, Cristina, M. Vincenta Ferrer, Salvador Hernández, and Boaz Tsaban. "The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory." Journal of Algebra 420 (December 2014): 86–119. http://dx.doi.org/10.1016/j.jalgebra.2014.06.040.

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38

Hegedűs, G. "WEAK REFLECTION IN CONCRETE CATEGORIES." Studia Scientiarum Mathematicarum Hungarica 37, no. 1-2 (2001): 185–93. http://dx.doi.org/10.1556/sscmath.37.2001.1-2.10.

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In this art cle I give the de .n t on of the strong and weak re .ex v ty and the com- pat bility relation of objects in a categorical language.These concepts will generalize the corresponding concepts n the theory of topolog cal vector spaces. The main theorem makes clear the connection between these mportant concepts and we can show a lot of objects in our category,which are not strongly re .exive,but compatible with strongly re .ex ve objects.We also consider a lot of interesting examples and try to throw new light upon the ex stence of such Abelian groups which satisfy Pontryagin duality but do not respect compactness.We w ll prove also that the weak topolog cal vector space- structures are not re .ex ve in general.
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39

Hofmann, Karl H., and Sidney A. Morris. "Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century." Axioms 10, no. 3 (2021): 190. http://dx.doi.org/10.3390/axioms10030190.

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This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.
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40

Oh, Yuna, and Jun Moon. "The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints." AIMS Mathematics 9, no. 3 (2024): 6109–44. http://dx.doi.org/10.3934/math.2024299.

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<abstract><p>In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.</p></abstract>
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41

JUNGE, MARIUS, MATTHIAS NEUFANG, and ZHONG-JIN RUAN. "A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS." International Journal of Mathematics 20, no. 03 (2009): 377–400. http://dx.doi.org/10.1142/s0129167x09005285.

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Recently, Neufang, Ruan and Spronk proved a completely isometric representation theorem for the measure algebra M(G) and for the completely bounded (Herz–Schur) multiplier algebra McbA(G) on [Formula: see text], where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups 𝔾 = (M, Γ, φ, ψ). More precisely, we introduce the algebra [Formula: see text] of completely bounded right multipliers on L1(𝔾) and we show that [Formula: see text] can be identified with the algebra of normal completely bounded [Formula: see text]-bimodule maps on [Formula: see text] which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L1(𝔾) is in fact implemented by an element of [Formula: see text]. We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutation relation.
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42

Hamaguchi, Yushi. "Infinite horizon backward stochastic Volterra integral equations and discounted control problems." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 101. http://dx.doi.org/10.1051/cocv/2021098.

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Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted L2-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted L2-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin’s maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.
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43

Geisser, Thomas H., and Baptiste Morin. "PONTRYAGIN DUALITY FOR VARIETIES OVER p-ADIC FIELDS." Journal of the Institute of Mathematics of Jussieu, September 28, 2022, 1–38. http://dx.doi.org/10.1017/s1474748022000469.

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Abstract We define cohomological complexes of locally compact abelian groups associated with varieties over p-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.
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44

Ayala, David, and John Francis. "ZERO-POINTED MANIFOLDS." Journal of the Institute of Mathematics of Jussieu, July 2, 2019, 1–74. http://dx.doi.org/10.1017/s1474748019000343.

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We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin–Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in${\mathcal{E}}_{n}$-algebra.
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45

Chavarria, Nicolas, and Anand Pillay. "A note on Pontryagin duality and continuous logic." Topology and its Applications, September 2022, 108260. http://dx.doi.org/10.1016/j.topol.2022.108260.

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46

Ferrer, María V., Julio Hernández-Arzusa, and Salvador Hernández. "Tensor products of topological abelian groups and Pontryagin duality." Journal of Mathematical Analysis and Applications, February 2024, 128199. http://dx.doi.org/10.1016/j.jmaa.2024.128199.

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47

McKee, Andrew. "Multipliers and Duality for Group Actions." Journal of Fourier Analysis and Applications 27, no. 6 (2021). http://dx.doi.org/10.1007/s00041-021-09893-4.

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AbstractWe define operator-valued Schur and Herz–Schur multipliers in terms of module actions, and show that the standard properties of these multipliers follow from well-known facts about these module actions and duality theory for group actions. These results are applied to study the Herz–Schur multipliers of an abelian group acting on its Pontryagin dual: it is shown that a natural subset of these Herz–Schur multipliers can be identified with the classical Herz–Schur multipliers of the direct product of the group with its dual group.
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48

JAMNESHAN, ASGAR, and TERENCE TAO. "An uncountable Moore–Schmidt theorem." Ergodic Theory and Dynamical Systems, May 11, 2022, 1–28. http://dx.doi.org/10.1017/etds.2022.36.

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Abstract We prove an extension of the Moore–Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a ‘conditional’ Pontryagin duality for spaces of abstract measurable maps.
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49

BENNETT, JONATHAN, and EUNHEE JEONG. "Fourier duality in the Brascamp–Lieb inequality." Mathematical Proceedings of the Cambridge Philosophical Society, September 27, 2021, 1–23. http://dx.doi.org/10.1017/s0305004121000608.

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Abstract It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.
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50

Lewis, Wayne, Peter Loth, and Adolf Mader. "The main decomposition of finite-dimensional protori." Journal of Group Theory, August 12, 2020. http://dx.doi.org/10.1515/jgth-2020-0079.

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AbstractA protorus is a compact connected abelian group of finite dimension. We use a result on finite-rank torsion-free abelian groups and Pontryagin duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus free complementary factor. We also classify direct products of protori of dimension 1 by means of canonical “type” subgroups. In addition, we produce the duals of some fundamental theorems of discrete abelian groups.
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