December  2019, 12(8): 2365-2377. 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, 2019, 12 (8) : 2365-2377. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[12]
[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

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

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[12]
[13]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

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

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

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