Pontryagin Duality and the Structure of Locally Compact Abelian Groups

Morris, Sidney A.

doi:10.1017/cbo9780511600722

Cited by 276 publications

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“…Since √ 2 and √ 3 are rationally independent, we can therefore apply Kronecker's lemma (Theorem 28 of [16]) and find k with…”

Section: Proof Letmentioning

confidence: 99%

On invariants for measure preserving transformations

Hjorth

2001

Fund. Math.

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Abstract. The classification problem for measure preserving transformations is strictly more complicated than that of graph isomorphism. Preamble.We consider the group M ∞ of all invertible measure preserving transformations either on the unit interval or any other reasonable measure space. It seems natural to say that two of these transformations, σ 1 , σ 2 , are equivalent or isomorphic if there is a third, π, so thatTo what extent can this equivalence relation be considered classifiable? In specific cases-for instance σ 1 , σ 2 both Bernoulli or discrete spectrum-there are well accepted systems of complete invariants. However, in the completely general context of arbitrary measure preserving transformations there is no known satisfactory system of complete invariants nor even a clear statement of what this would entail.For instance, Halmos in [7] despairs of precisely formulating the problem but at page 1029 suggests that its solution should fulfill the "the vague task of finding a complete set of invariants. . . " At page 75 of [8] he proposes that the central problem is to "find usable necessary and sufficient conditions for the conjugacy of two measure preserving transformations." Some time later Weiss at page 670 of [23] raises the problem of finding "a set of invariants large enough so that if all invariants agree for two m.p.t. one can conclude that the m.p.t. are isomorphic".This article considers attempts to make this precise and ask abstractly whether the problem of classifiability could in principle have a positive solu-2000 Mathematics Subject Classification: Primary 03A15. Key words and phrases: classification, measure preserving transformation, Polish group action.The author gratefully acknowledges support from NSF grants DMS 99-70403 and DMS 96-22977.[51] 52 G. Hjorth tion. We present a clearly identifiable lower bound on the classification difficulty of the isomorphism relation for measure preserving transformations.One precise formulation of the problem would be to understand a classifiable equivalence relation on a Polish space to be one for which we may find a Borel assignment of reals or points in some other standard Borel space as complete invariants. This is the notion of classifiable suggested by the Glimm-Effros dichotomy of [10]. Indeed, Feldman in [4] takes exactly that position. Appealing to [18] he observes that the isomorphism relation for Bernoulli shifts allows real numbers to be assigned in a Borel manner as a complete invariant and uses [19] to remark that such an assignment is already impossible for the class of measure preserving transformations having the property of K.A more generous notion of classification, closer to the kinds we consider below, is already implicit in sources [7], [8], [23]. In each case the results of [9] are accepted as providing a complete classification for the discrete spectrum measure preserving transformations. Here the invariants are not real numbers or single points in a standard Borel space, but rather countable sets of complex numbers. The sig...

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“…Since √ 2 and √ 3 are rationally independent, we can therefore apply Kronecker's lemma (Theorem 28 of [16]) and find k with…”

Section: Proof Letmentioning

confidence: 99%

On invariants for measure preserving transformations

Hjorth

2001

Fund. Math.

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“…(This wellknown result follows easily for example from Corollary 1 of Theorem 14 of [5].) We shall let re, denote the projection of G onto the factor G h so that n i = p Proof.…”

Section: Let / Be a Canonical Continuous Homomorphism Of T M Onto T"mentioning

confidence: 96%

A note an homeomorphic measures on topological groups

Morris

Peck

1983

Proceedings of the Edinburgh Mathematical Society

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The classical von Neumann–Oxtoby–Ulam Theorem states the following:Given non-atomic Borel probability measures μ, λ on In such thatthere exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set SIt is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].

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“…χ(x + y) = χ(x)χ(y), |χ(x)| = 1, see [11] for a general theory). The Pontryagin dual group A can be identified with A by the homeomorphism s → χ s given by χ s (t) = χ(st).…”

Section: The Linear Theorymentioning

confidence: 99%

Generalized functions on adeles. Linear and non-linear theories

Radyno

Radyna

2010

Linear and Non-Linear Theory of Generalized Functions and Its Applications

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Abstract. We consider various generalizations of linear homogeneous distributions on adeles and construct a number of algebras of non-linear generalized functions on adeles and totally disconnected groups such as the discrete adeles.1. Introduction. The algebra of adeles over the field of rational numbers Q was introduced by A. Weil. Invertible adeles, now known as ideles, were introduced by C. Chevalley [4]. Adeles became a powerful analytic tool in number theory [20]; they have applications in representation theory [9] and in modern mathematical physics [6].There is no interesting topology on the field of rational numbers. However it is possible to embed this field into the canonically associated algebra of adeles having a non-trivial topology and a Haar measure. Thus numerous tools of functional analysis can be applied. Here we summarize some developments in linear and non-linear theory of generalized functions on adeles. See [5] for basics of the non-linear theory on the real axis. An abstract approach to the non-linear theory is presented in [2,15].

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Pontryagin Duality and the Structure of Locally Compact Abelian Groups

Cited by 276 publications

References 0 publications

On invariants for measure preserving transformations

On invariants for measure preserving transformations

A note an homeomorphic measures on topological groups

Generalized functions on adeles. Linear and non-linear theories

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