Transformation Groups and C ∗ -algebras

Effros, Edward G.

doi:10.2307/1970381

Cited by 316 publications

(164 citation statements)

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“…In this context one could compare the conjugacy equivalence relation on the unitary representations of a discrete group: in the case of irreducible representations this is known to be not only Borel but actually F σ in an appropriate topology. (See [3]. )…”

Section: Preamblementioning

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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Section: Preamblementioning

confidence: 99%

On invariants for measure preserving transformations

Hjorth

2001

Fund. Math.

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“…The sets U^, and U60 are disjoint; for otherwise, by (9) and (10), there exists a circular chain in 60 U S,, and this contradicts the fact that 9" is a tree chain. Hence 6D,\eo^0.…”

Section: Definitions and Related Results A Space Is Homogeneous If Fmentioning

confidence: 96%

No Homogeneous Tree-Like Continuum Contains an Arc

Hagopian¹

1983

Proceedings of the American Mathematical Society

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“…The main tool of this paper is Theorem 2.1 of [5]. This theorem gives a condition for a space to be a coset space of a topological group, and as a direct result of it, we can prove that compact metric homogeneous spaces and locally compact locally connected separable metric spaces are cosets spaces of their group of homeomorphisms with the C -0 topology.…”

mentioning

confidence: 96%

“…Everything else. We are now in a position to state Theorem 2.1 of [5]. Let (G, X) be a polonais transformation group.…”

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confidence: 99%

On All Kinds of Homogeneous Spaces

Ungar¹

1975

Transactions of the American Mathematical Society

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ABSTRACT. Several open questions on homogeneous spaces are answered.A few of the results are:(1) An «-homogeneous metric continuum, which is not the circle, is strongly «-homogeneous.(2) A 2-homogeneous metric continuum is locally connected.(3) If A1 is a homogeneous compact metric space or a homogeneous locally compact, locally connected separable metric space, then X is a coset space.(4) If G is a complete separable metric topological group with is «-connected, then G is locally «-connected.1. Introduction. At the topology conference at the University of Wisconsin in 1955 Burgess [3] asked the following questions.(1) Does there exist a continuum different from a simple closed curve which, for some positive integer n, is «-homogeneous but not strongly «-homogeneous?(2) Is every 2-homogeneous compact metric continuum locally connected? (3) Are there continua for which to each pair of their points there exists a unique homeomorphism carrying one of the points onto the other?The paper will give almost complete answers to all of these questions in the metric case (although only (2) mentions the word metric, all of Burgess' work on homogeneity is on metric spaces). The answer to (1) will be no. The answer to (2) will be yes. It will also be shown that a strong type of 1-homogeneity gives an affirmative answer to (2). It seems like the answer to (3) should be no. It is shown that if a space satisfies the condition of (3) then it is an abelian topological group and applying structure theorems in [9], it is shown that it cannot be finite dimensional.The main tool of this paper is Theorem 2.1 of [5]. This theorem gives a condition for a space to be a coset space of a topological group, and as a direct result of it, we can prove that compact metric homogeneous spaces and locally compact locally connected separable metric spaces are cosets spaces of their group of homeomorphisms with the C -0 topology. This answers questions raised by the work of Ford [6], Mostert [10] and Ungar [12]. As a matter of fact in these last three references, two types of homogeneity were defined in order to

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Transformation Groups and C ∗ -algebras

Cited by 316 publications

References 16 publications

On invariants for measure preserving transformations

On invariants for measure preserving transformations

No Homogeneous Tree-Like Continuum Contains an Arc

On All Kinds of Homogeneous Spaces

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