doi: 10.3934/dcdss.2019148

On a semigroup problem

1. 

Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, USA

2. 

Institute of Mathematics, Romanian Academy, P.O. Box 1-764, RO-70700 Bucharest, Romania

3. 

Department of Mathematics, University of Houston, Houston, TX 77204-3308, USA

* Corresponding author: Viorel Nitica

Received  July 2016 Revised  October 2017 Published  January 2019

Fund Project: VN was partially supported by Simons Foundation Grant 208729. AT was partially supported by Simons Foundation Grant 239583

If $ S $ is a semigroup in $ \mathbb{R}^n $ that is not separated by a linear functional, then it is known that the closure of $ S $ is a group. We investigate a similar statement in an infinite dimensional topological vector space $ X $. We show that if $ X $ is an infinite dimensional Banach space, then there exists a semigroup $ S\subset X $, not separated by the continuous functionals supported by the closed linear span of $ S $, for which the closure of the semigroup is not a group. If $ X $ is an infinite dimensional Fréchet space, then the closure of a semigroup that is not separated is always a group if and only if $ X $ is $ \mathbb{R}^{\omega} $, the countably infinite direct product of lines. Other infinite dimensional topological vector spaces, such as $ \mathbb{R}^{\infty} $, the countably infinite direct sum of lines, are discussed. The Semigroup Problem has applications to the study of certain dynamical systems, in particular for the construction of topologically transitive extensions of hyperbolic systems. Some examples are shown in the paper.

Citation: Viorel Nitica, Andrei Török. On a semigroup problem. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019148
References:
[1]

C. R. Adams, The space of functions of bounded variation and certain general spaces, Trans. Amer. Math. Soc., 40 (1936), 421-438. doi: 10.1090/S0002-9947-1936-1501882-8.

[2]

R. D. Anderson, Hibert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 72 (1966), 515-519. doi: 10.1090/S0002-9904-1966-11524-0.

[3]

C. Bargetz, Completing the Valdivia-Vogt tables of sequence-space representations of spaces of smooth functions and distributions, Monatshefte für Mathematik, 177 (2015), 1-14. doi: 10.1007/s00605-014-0650-2.

[4]

C. BessagaA. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of $F$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 9 (1961), 677-683.

[5]

P. A. Borodin, Density of a semigroup in a Banach space, Izvestiya: Mathematics, 78 (2014), 1079-1104. doi: 10.1070/im2014v078n06abeh002721.

[6]

R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35. American Mathematical Society Providence, R.I. 1978.

[7]

R. BrownP. J. Higgins and S. A. Morris, Countable products and sums of lines and circles: Their closed subgroups, quotients and duality properties, Math. Proc. Camb. Phil. Soc., 78 (1975), 19-32. doi: 10.1017/S0305004100051483.

[8]

H. Jarchow, Locally Convex Spaces, Springer, 1981.

[9]

N. Kalton, Normalization properties of Schauder bases, Proc. London Math.. Soc., 22 (1971), 91-105. doi: 10.1112/plms/s3-22.1.91.

[10]

N. Kalton, The metric linear spaces $ L_p $ for $ 0<p<1 $, Contemporary Mathematics, 52 (1986), 55-69. doi: 10.1090/conm/052/840695.

[11]

N. Kalton, The basic sequence problem, Studia Mathematica, 116 (1995), 168-187. doi: 10.4064/sm-116-2-167-187.

[12]
[13]

J. Lindestrauss and L. Tzafriri, Classic Banach Spaces Ⅰ, Ⅱ, Springer-Verlag, 1977.

[14]

K. LuiV. Nitica and S. Venkatesh, The semigroup problem for central semidirect product of $ \mathbb{R} ^n $ with $ \mathbb{R} ^m $, Topology Proceedings, 45 (2015), 9-29.

[15]

P. Mankiewicz, On topological, Lipschitz, and uniform classification of LF-spaces, Studia Math., 54 (1974), 109-142. doi: 10.4064/sm-52-2-109-142.

[16]

I. MelbourneV. Nitica and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems, Annales Henri Poincaré, 6 (2005), 725-746. doi: 10.1007/s00023-005-0221-0.

[17]

I. MelbourneV. Nitica and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems, Contin. Dynam. Systems, 14 (2006), 355-363. doi: 10.3934/dcds.2006.14.355.

[18]

S. A. Morris, Locally compact abelian groups and the variety of topological groups generated by the reals, Proc. Amer. Math. Soc., 34 (1972), 290-292. doi: 10.1090/S0002-9939-1972-0294560-4.

[19]

V. Nitica and A. Török, Open and dense topological transitivity of extensions by non-compact fiber of hyperbolic systems: a review, Axioms, 4 (2015), 84-101.

[20]

V. Nitica and A. Török, Stable transitivity of Heisenberg group extensions of hyperbolic systems, Nonlinearity, 27 (2014), 661-683. doi: 10.1088/0951-7715/27/4/661.

[21]

V. Nitica and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 257-269. doi: 10.1017/S0143385704000471.

[22]

A. Pietsch, Nuclear Locally Convex Spaces, Ergebnisse Der Mathematick und Ihrer Grensgebiete, Volume 66, Springer-Verlag, New York/Heidelberg/Berlin, 1972.

[23]

Z. Rosengarten and A. Reich, Transitivity of infinite dimensional extensions of Anosov diffeomorphisms, 2012, arXiv: 1209.2183v1.

[24]

W. Rudin, Functional Analysis, second edition, McGraw-Hill, 1991.

[25]

J. H. Shapiro, On the weak basis theorem in $F$-spaces, Canadian J. Math., 26 (1974), 1294-1300. doi: 10.4153/CJM-1974-124-5.

[26]

E. A. Sidorov, Topologically transitive cylindrical cascades (Russian), Mat. Zametki, 14 (1973), 441-452.

[27]

T. Silverman and S. M. Miller, Hilbert extensions of Anosov diffeomorpisms, 2013, http://www.math.psu.edu/mass/reu/2013/mathfest/HilbertCounterexample.pdf

[28] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.
[29]

J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Mathematica, 120 (19966), 247-258. doi: 10.4064/sm-120-3-247-258.

show all references

References:
[1]

C. R. Adams, The space of functions of bounded variation and certain general spaces, Trans. Amer. Math. Soc., 40 (1936), 421-438. doi: 10.1090/S0002-9947-1936-1501882-8.

[2]

R. D. Anderson, Hibert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 72 (1966), 515-519. doi: 10.1090/S0002-9904-1966-11524-0.

[3]

C. Bargetz, Completing the Valdivia-Vogt tables of sequence-space representations of spaces of smooth functions and distributions, Monatshefte für Mathematik, 177 (2015), 1-14. doi: 10.1007/s00605-014-0650-2.

[4]

C. BessagaA. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of $F$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 9 (1961), 677-683.

[5]

P. A. Borodin, Density of a semigroup in a Banach space, Izvestiya: Mathematics, 78 (2014), 1079-1104. doi: 10.1070/im2014v078n06abeh002721.

[6]

R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35. American Mathematical Society Providence, R.I. 1978.

[7]

R. BrownP. J. Higgins and S. A. Morris, Countable products and sums of lines and circles: Their closed subgroups, quotients and duality properties, Math. Proc. Camb. Phil. Soc., 78 (1975), 19-32. doi: 10.1017/S0305004100051483.

[8]

H. Jarchow, Locally Convex Spaces, Springer, 1981.

[9]

N. Kalton, Normalization properties of Schauder bases, Proc. London Math.. Soc., 22 (1971), 91-105. doi: 10.1112/plms/s3-22.1.91.

[10]

N. Kalton, The metric linear spaces $ L_p $ for $ 0<p<1 $, Contemporary Mathematics, 52 (1986), 55-69. doi: 10.1090/conm/052/840695.

[11]

N. Kalton, The basic sequence problem, Studia Mathematica, 116 (1995), 168-187. doi: 10.4064/sm-116-2-167-187.

[12]
[13]

J. Lindestrauss and L. Tzafriri, Classic Banach Spaces Ⅰ, Ⅱ, Springer-Verlag, 1977.

[14]

K. LuiV. Nitica and S. Venkatesh, The semigroup problem for central semidirect product of $ \mathbb{R} ^n $ with $ \mathbb{R} ^m $, Topology Proceedings, 45 (2015), 9-29.

[15]

P. Mankiewicz, On topological, Lipschitz, and uniform classification of LF-spaces, Studia Math., 54 (1974), 109-142. doi: 10.4064/sm-52-2-109-142.

[16]

I. MelbourneV. Nitica and A. Török, Stable transitivity of certain noncompact extensions of hyperbolic systems, Annales Henri Poincaré, 6 (2005), 725-746. doi: 10.1007/s00023-005-0221-0.

[17]

I. MelbourneV. Nitica and A. Török, A note about stable transitivity of noncompact extensions of hyperbolic systems, Contin. Dynam. Systems, 14 (2006), 355-363. doi: 10.3934/dcds.2006.14.355.

[18]

S. A. Morris, Locally compact abelian groups and the variety of topological groups generated by the reals, Proc. Amer. Math. Soc., 34 (1972), 290-292. doi: 10.1090/S0002-9939-1972-0294560-4.

[19]

V. Nitica and A. Török, Open and dense topological transitivity of extensions by non-compact fiber of hyperbolic systems: a review, Axioms, 4 (2015), 84-101.

[20]

V. Nitica and A. Török, Stable transitivity of Heisenberg group extensions of hyperbolic systems, Nonlinearity, 27 (2014), 661-683. doi: 10.1088/0951-7715/27/4/661.

[21]

V. Nitica and M. Pollicott, Transitivity of Euclidean extensions of Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 25 (2005), 257-269. doi: 10.1017/S0143385704000471.

[22]

A. Pietsch, Nuclear Locally Convex Spaces, Ergebnisse Der Mathematick und Ihrer Grensgebiete, Volume 66, Springer-Verlag, New York/Heidelberg/Berlin, 1972.

[23]

Z. Rosengarten and A. Reich, Transitivity of infinite dimensional extensions of Anosov diffeomorphisms, 2012, arXiv: 1209.2183v1.

[24]

W. Rudin, Functional Analysis, second edition, McGraw-Hill, 1991.

[25]

J. H. Shapiro, On the weak basis theorem in $F$-spaces, Canadian J. Math., 26 (1974), 1294-1300. doi: 10.4153/CJM-1974-124-5.

[26]

E. A. Sidorov, Topologically transitive cylindrical cascades (Russian), Mat. Zametki, 14 (1973), 441-452.

[27]

T. Silverman and S. M. Miller, Hilbert extensions of Anosov diffeomorpisms, 2013, http://www.math.psu.edu/mass/reu/2013/mathfest/HilbertCounterexample.pdf

[28] F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.
[29]

J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Mathematica, 120 (19966), 247-258. doi: 10.4064/sm-120-3-247-258.

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