Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-11T14:54:01.492Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE COMPLEXITY OF CLASSIFYING LEBESGUE SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices Computability and recursion theory

Published online by Cambridge University Press:  26 October 2020

TYLER A. BROWN ,
TIMOTHY H. MCNICHOLL  and
ALEXANDER G. MELNIKOV
TYLER A. BROWN
Affiliation:
DEPARTMENT OF MATHEMATICS IOWA STATE UNIVERSITYAMES, IOWA50011, USA DIVISION OF HEALTH, MATHEMATICS, AND SCIENCES COLUMBIA COLLEGECOLUMBIA, SOUTH CAROLINA29203, USAE-mail: [email protected]
TIMOTHY H. MCNICHOLL
Affiliation:
DEPARTMENT OF MATHEMATICS IOWA STATE UNIVERSITYAMES, IOWA50011, USAE-mail: [email protected]
ALEXANDER G. MELNIKOV
Affiliation:
THE INSTITUTE OF NATURAL AND MATHEMATICAL SCIENCES PRIVATE BAG 102 904 NSMC ALBANY 0745, AUCKLAND, NEW ZEALANDE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.

Keywords

computable analysiscomputable structure theoryLpspaces

MSC classification

Primary: 03D45: Theory of numerations, effectively presented structures
Secondary: 03D78: Computation over the reals 46B04: Isometric theory of Banach spaces
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 3 , September 2020 , pp. 1254 - 1288
Check for updates
Copyright
© The Association for Symbolic Logic 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Part of this research was conducted while T. McNicholl visited A. Melnikov. This visit was funded by Marsden Fund of New Zealand and by Simons Foundation Grant # 317870.

References

REFERENCES

Ash, C. and Knight, J., Computable Structures and the Hyperarithmetical Hierarchy , Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.Google Scholar
Brattka, V., Hertling, P., and Weihrauch, K., A tutorial on computable analysis . New Computational Paradigms (Cooper, B. S., Löwe, B., and Sorbi, A., editors), Springer, New York, NY, 2008, pp. 425–491.CrossRefGoogle Scholar
Brown, T. and McNicholl, T. H., Analytic computable structure theory and ${L}^p$ -spaces part 2, submitted. Preprint. 2018. Available at http://arxiv.org/abs/1801.00355.Google Scholar
Calvert, W., Harizanov, V., Knight, J., and Miller, S., Index sets of computable models . Algebra Logika , vol. 45 (2006), no. 5, pp. 538–574, 631–632.CrossRefGoogle Scholar
Carson, J., Harizanov, V., Knight, J., Lange, K., McCoy, C., Morozov, A., Quinn, S., Safranski, C., and Wallbaum, J., Describing free groups . Transactions of the American Mathematical Society , vol. 364 (2012), no. 11, pp. 5715–5728.CrossRefGoogle Scholar
Cembranos, P. and Mendoza, J., Banach Spaces of Vector-Valued Functions , Lecture Notes in Mathematics, vol. 1676, Springer-Verlag, Berlin, 1997.CrossRefGoogle Scholar
Cenzer, D. and Remmel, J. B., Index sets in computable analysis . Theoretical Computer Science , vol. 219 (1999), no. 1-2, pp. 111–150, Computability and complexity in analysis (Castle Dagstuhl, 1997).CrossRefGoogle Scholar
Clanin, J., McNicholl, T. H., and Stull, D. M., Analytic computable structure theory and ${L}^p$ spaces . Fundamenta Mathematicae , vol. 244 (2019), no. 3, pp. 255–285.CrossRefGoogle Scholar
Downey, R. and Melnikov, A., Effectively categorical abelian groups . Journal of Algebra , vol. 373 (2013), pp. 223–248.CrossRefGoogle Scholar
Downey, R. and Montalbán, A., The isomorphism problem for torsion-free Abelian groups is analytic complete , Journal of Algebra , vol. 320 (2008), no. 6, pp. 2291–2300.CrossRefGoogle Scholar
Downey, R. and Melnikov, A. G., Computable completely decomposable groups . Transactions of the American Mathematical Society , vol. 366 (2014), no. 8, pp. 4243–4266.CrossRefGoogle Scholar
Downey, R. G., Melnikov, A. G., and Ng, K. M., A Friedberg enumeration of equivalence structures . Journal of Mathematical Logic , vol. 17 (2017), no. 2, p 1750008, 28.CrossRefGoogle Scholar
Epstein, R. L., Haas, R., and Kramer, R. L., Hierarchies of sets and degrees below 0, Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80) (Lerman, M., Schmerl, J.H., and Soare, R.I., editors), Lecture Notes in Mathematics, vol. 859, Springer, Berlin, 1981, pp. 32–48.Google Scholar
Ershov, Y. and Goncharov, S., Constructive Models , Siberian School of Algebra and Logic, Consultants Bureau, New York, NY, 2000.CrossRefGoogle Scholar
Ershov, Y. L., A hierarchy of sets, Part I . Algebra and Logic , vol. 7 (1968), pp. 24–43, (English translation).Google Scholar
Ershov, Y. L., A hierarchy of sets, Part II . Algebra and Logic , vol. 7 (1968), pp. 212–232, (English translation).CrossRefGoogle Scholar
Ershov, Y. L., A hierarchy of sets, Part III . Algebra and Logic , vol. 9 (1970), pp. 20–31, (English translation).CrossRefGoogle Scholar
Fuchs, L., Infinite Abelian Groups . Volume I, Pure and Applied Mathematics, vol. 36, Academic Press, New York, NY, 1970.Google Scholar
Fuchs, L.. Infinite Abelian Groups . Volume II, Pure and Applied Mathematics, vol. 36, Academic Press, New York, NY, 1973.Google Scholar
Goncharov, S., Countable Boolean Algebras and Decidability , Siberian School of Algebra and Logic, Consultants Bureau, New York, 1997.Google Scholar
Goncharov, S. and Knight, J., Computable structure and antistructure theorems . Algebra Logika , vol. 41 (2002), no. 6, pp. 639–681, 757.Google Scholar
Greenberg, N., Turetsky, D., and Westrick, L. B., Finding bases of uncountable free abelian groups is usually difficult . Transactions of the American Mathematical Society , vol. 370 (2018), no. 6, pp. 4483–4508.CrossRefGoogle Scholar
Halmos, P. R., Measure Theory , D. Van Nostrand, New York, 1950.CrossRefGoogle Scholar
Hanner, O., On the uniform convexity of Lp and lp . Arkiv för Matematik , vol. 3 (1956), pp. 239–244.CrossRefGoogle Scholar
Kakutani, S., Concrete representation of abstract $(L)$ -spaces and the mean ergodic theorem, Annals of Mathematics , vol. 42 (1941), 2, pp. 523–537.CrossRefGoogle Scholar
Lacey, H. E., The Isometric Theory of Classical Banach Spaces , Die Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974.CrossRefGoogle Scholar
Lamperti, J., On the isometries of certain function-spaces . Pacific Journal of Mathematics , vol. 8 (1958), pp. 459–466.CrossRefGoogle Scholar
Mal’cev, A., Constructive algebras. I . Uspekhi Matematicheskikh Nauk , vol. 16 (1961), no. 3 (99), pp. 3–60.Google Scholar
McCoy, C. and Wallbaum, J., Describing free groups, Part II: ${\varPi}_4^0$ hardness and no ${\varSigma}_2^0$ basis. Transactions of the American Mathematical Society , vol. 364 (2012), no. 11, pp. 5729–5734.CrossRefGoogle Scholar
McNicholl, T. H., Computable copies of ℓp . Computability , vol. 6 (2017), no. 4, pp. 391–408.CrossRefGoogle Scholar
McNicholl, T. H., Continuous logic and embeddings of Lebesgue spaces. Mathematical Logic Quarterly , to appear. Preprint. 2019. Available at http://arxiv.org/abs/1805.11561.Google Scholar
McNicholl, T. H., Computing the exponent of a Lebesgue space, Submitted. Preprint. 2020. Available at https://arxiv.org/abs/1912.13087.CrossRefGoogle Scholar
McNicholl, T. H. and Stull, D. M., The isometry degree of a computable copy of ${\ell}^p$ , Computability , to appear, 2019. Available at https://content.iospress.com/articles/computability/com180214.CrossRefGoogle Scholar
Melnikov, A., Computable topological groups and Pontryagin duality . Transactions of the American Mathematical Society , vol. 370 (2018), no. 12, pp. 8709–8737.CrossRefGoogle Scholar
Melnikov, A. and Montalbán, A., Computable Polish group actions , this Journal , vol. 83 (2018), no. 2, pp. 443–460.Google Scholar
Melnikov, A. G., Computably isometric spaces , this Journal , vol. 78 (2013), no. 4, pp. 1055–1085.Google Scholar
Melnikov, A. G. and Ng, K. M., Computable structures and operations on the space of continuous functions . Fundamenta Mathematicae , vol. 233 (2016), no. 2, pp. 101–141.Google Scholar
Melnikov, A. G. and Nies, A., The classification problem for compact computable metric spaces . The Nature of Computation (Bonizzoni, P., Brattka, V. and Löwe, B., editors), Lecture Notes in Computer Science, vol. 7921, Springer, Heidelberg, 2013, pp. 320–328.Google Scholar
Nies, A. and Solecki, S., Local compactness for computable polish metric spaces is ${\varPi}_1^1$ -complete, Evolving Computability - 11th Conference on Computability in Europe, CIE 2015 , 2015, pp. 286–290.CrossRefGoogle Scholar
Pour-El, M. B. and Richards, J. I., Computability in Analysis and Physics , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989.CrossRefGoogle Scholar
Rabin, M., Computable algebra, general theory and theory of computable fields . Transactions of the American Mathematical Society , vol. 95 (1960), pp. 341–360.Google Scholar
Riggs, K., The decomposability problem for torsion-free abelian groups is analytic-complete . Proceedings of the American Mathematical Society , vol. 143 (2015), no. 8, pp. 3631–3640.CrossRefGoogle Scholar
Rogers, H., Theory of Recursive Functions and Effective Computability , second ed., MIT Press, Cambridge, MA, 1987.Google Scholar
Rogers, L., Ulm’s theorem for partially ordered structures related to simply presented Abelian $p$ -groups , Transactions of the American Mathematical Society , vol. 227 (1977), pp. 333–343.Google Scholar
Soare, R., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem . Proceedings of the London Mathematical Society , vol. 42 (1936), pp. 230–265.Google Scholar
Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem. A correction . Proceedings of the London Mathematical Society , vol. 43 (1937), pp. 544–546.Google Scholar
Weihrauch, K., Computable Analysis: An Introduction , Texts in Theoretical Computer Science. Springer-Verlag, Berlin, 2000.CrossRefGoogle Scholar