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Actions by the classical Banach spaces

  • G. Hjorth (a1)

Extract

The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.

For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.

This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that the Borel cardinalities of the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.

One of the first things to be addressed concerns the complexity of these equivalence relations. This question for appears in [1].

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References

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[1]Becker, H. and Kechris, A. S., The descriptive set theory of Polish group actions, to appear in the London Mathematical Society Lecture Notes Series.
[2]Dougherty, R. and Hjorth, G., Reducibility and nonreducibility between lp equivalence relations, to appear in the Transactions of the American Mathematical Society.
[3]Effros, E., Transformation groups and C*-algebras, Annals of Mathematics, vol. 81 (1975), pp. 3855.
[4]Farah, I., Basis problem for turbulent actions, preprint, York University, Toronto, 1998.
[5]Feldman, J., Hahn, P., and Moore, C. C., Orbit structure and countable sections for actions of countable groups, Advances in Mathematics, vol. 28 (1978), pp. 186230.
[6]Harrington, L. A., A powerless proof of a theorem by Silver, handwritten note, UC Berkeley, 1976.
[7]Harrington, L. A., Kechris, A. S., and Louveau, A., A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), pp. 903928.
[8]Harrington, L. A., Marker, D., and Shelah, S., Borel orderings, Transactions of the American Mathematical Society, vol. 310 (1988), pp. 292302.
[9]Hjorth, G., An absoluteness principle for Borel sets, preprint, Caltech, 1996.
[10]Hjorth, G., Classification and orbit equivalence relations, manuscript, UCLA, 1997.
[11]Hjorth, G., Kechris, A. S., and Louveau, A., Borel equivalence relations induced by actions of the symmetric group, preprint, Caltech, 1996.
[12]Hodges, W., Model theory, Cambridge University Press, Cambridge, 1993.
[13]Jackson, S., Kechris, A. S., and Louveau, A., On countable Borel equivalence relations, preprint, Caltech, 1994.
[14]Jech, T., Set theory, Academic Press, New York, 1978.
[15]Kechris, A. S., Definable equivalence relations and Polish group actions, unpublished manuscript.
[16]Kechris, A. S., Countable ordinals and the analytic hierarchy, II, Annals of Mathematical Logic, vol. 15 (1978), pp. 193223.
[17]Kechris, A. S., The structure of Borel equivalence relations in Polish spaces, Set theory of the continuum (Judah, H., Just, W., and Woodin, W. H., editors), MSRI Publication, no. 26, Springer-Verlag, New York, 1990, pp. 7584.
[18]Kechris, A. S., Countable sections for locally compact group actions, Journal of Ergodic Theory and Dynamical Systems, vol. 12 (1992), pp. 283295.
[19]Kechris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1995.
[20]Kechris, A. S., Notes on turbulence, typewritten notes, Caltech, 1996.
[21]Kechris, A. S., Rigidity properties of Borel ideals on the integers, in the 8th Prague Topological Symposium on general topology and its relations to modern analysis and algebra (1996), Topology and its Applications, vol. 85 (1998), pp. 195205.
[22]Kechris, A. S. and Louveau, A., The classification of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society, vol. 10 (1997), pp. 215242.
[23]Louveau, A. and Velickovic, B., A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), pp. 255259.
[24]Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam, 1980.
[25]Silver, J., Counting the number of equivalence classes of Borel and co-analytic equivalence relation, Annals of Mathematical Logic, vol. 18 (1980), pp. 128.
[26]Solecki, S., dissertation, Caltech, 1995.
[27]Solecki, S., Analytic ideals, Bulletin of Symbolic Logic, vol. 2 (1996), pp. 339348.
[28]Solecki, S., Actions of non-compact and non-locally compact Polish groups, preprint, University of Indiana, Bloomington, 1998.
[29]Zimmer, R., Ergodic theory and semi-simple groups, BirkhÄuser, Basel, 1984.

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  • G. Hjorth (a1)

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