# American Institute of Mathematical Sciences

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
 [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. Bessaga, A. 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. Brown, P. 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 $0 show all references ##### References:  [1] C. R. Adams, The space of functions of bounded variation and certain general spaces, Trans. Amer. Math. Soc., 40 (1936), 421-438. doi: 10.1090/S0002-9947-1936-1501882-8. Google Scholar [2] R. D. Anderson, Hibert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 72 (1966), 515-519. doi: 10.1090/S0002-9904-1966-11524-0. Google Scholar [3] C. Bargetz, Completing the Valdivia-Vogt tables of sequence-space representations of spaces of smooth functions and distributions, Monatshefte für Mathematik, 177 (2015), 1-14. doi: 10.1007/s00605-014-0650-2. Google Scholar [4] C. Bessaga, A. Pełczyński and S. Rolewicz, On diametral approximative dimension and linear homogeneity of$F$-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 9 (1961), 677-683. Google Scholar [5] P. A. Borodin, Density of a semigroup in a Banach space, Izvestiya: Mathematics, 78 (2014), 1079-1104. doi: 10.1070/im2014v078n06abeh002721. Google Scholar [6] R. Bowen, On Axiom A Diffeomorphisms, Regional Conference Series in Mathematics, No. 35. American Mathematical Society Providence, R.I. 1978. Google Scholar [7] R. Brown, P. J. Higgins and S. A. Morris, Countable products and sums of lines and circles: Their closed subgroups, quotients and duality properties, Math. Proc. Camb. Phil. Soc., 78 (1975), 19-32. doi: 10.1017/S0305004100051483. Google Scholar [8] H. Jarchow, Locally Convex Spaces, Springer, 1981. Google Scholar [9] N. Kalton, Normalization properties of Schauder bases, Proc. London Math.. Soc., 22 (1971), 91-105. doi: 10.1112/plms/s3-22.1.91. Google Scholar [10] N. Kalton, The metric linear spaces$ L_p $for$ 0
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