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Hilbert’s fifth problem for local groups. (English) Zbl 1219.22004
The author has started two versions of Hilbert’s fifth problem written below after some definitions:
(a) A local group is a tuple \((G,1,\iota,t, p)\) (or simply, \(G\) in short) where \(G\) is a Hausdorff topological space with a distinguished element \(1\in G\), an \(\iota:\Lambda\to G\) (the inversion map) and \(p:\Omega\to G\) (the product map) are continuous functions with \(\Lambda\subseteq G\) and open \(\Omega\subseteq G\times G\), such that \(1\in\Lambda\), \(\{1\}\times G\subseteq\Omega\), \(G\times \{1\}\subseteq\Omega\), and for all \(x,y,z\in G\):
(1) \(p(1, x)= p(x, 1)= x\); (2) if \(x\in\Lambda\), then \((x, \iota(x))\in\Omega\), \((\iota(x), x)\in\Omega\) and \(p(x,\iota(x))= p(\iota(x), x)= 1\); (3) if \((x, y), (y, z)\in\Omega\) and \((p(x, y),z)\), \((x, p(y,z))\in\Omega\), then (A) \(p(p(x,y),z)= p(x,p(y,z))\);
(b) \(G\) is locally Euclidean if there is an open neighbourhood of 1 homeomorphic to an open subset of \(\mathbb{R}^n\) for some \(n\);
(c) \(G\) is a local Lie group if \(G\) admits a \(C^\infty\) structure such that the maps \(\iota\) and \(p\) are \(C^w\);
(d) with \(U\) an open neighbourhood of 1 in \(G\), the restriction of \(G\) to \(U\) is the local group \(G/U:= (U,\iota/\Lambda_U, p/\Omega_U)\), where \(\Lambda_U:= \Lambda\cap U\cap \iota^{-1}(U)\) and \(\Omega_U:= (U\times U)\cap p^{-1}(U)\);
(e) \(G\) is globalizable if there is a topological group \(H\) and an open neighbourhood \(U\) of \(1_H\) in \(H\) such that \(G= H/U\).
Local \(H5\) first form: If \(G\) is a locally Euclidean local group, then some restriction of \(G\) is a local Lie group. Local \(H5\) second form: If \(G\) is a locally Euclidean local group, then some restriction of \(G\) is globalizable.
The equivalence of the two forms is established. The author follows the non-standard treatment of Hilbert’s fifth problem given by Hirschfeld; in dealing with local \(H5\), the author has a simplified way in the non-standard method.

MSC:
22A30 Other topological algebraic systems and their representations
22E99 Lie groups
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References:
[1] D. R. Brown and R. S. Houston, ”Cancellative semigroups on manifolds,” Semigroup Forum, vol. 35, iss. 3, pp. 279-302, 1987. · Zbl 0626.22001
[2] M. Davis, Applied Nonstandard Snalysis, New York: Wiley-Interscience [John Wiley & Sons], 1977. · Zbl 0359.02060
[3] L. van den Dries, Unpublished notes, 1981.
[4] A. M. Gleason, ”Groups without small subgroups,” Ann. of Math., vol. 56, pp. 193-212, 1952. · Zbl 0049.30105
[5] Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions; Nonstandard Analysis: Theory and ApplicationsDordrecht: Kluwer Academic Publishers Group, 1997. · Zbl 0910.03040
[6] J. Hirschfeld, ”The nonstandard treatment of Hilbert’s fifth problem,” Trans. Amer. Math. Soc., vol. 321, iss. 1, pp. 379-400, 1990. · Zbl 0705.03033
[7] K. H. Hofmann and W. Weiss, ”More on cancellative semigroups on manifolds,” Semigroup Forum, vol. 37, iss. 1, pp. 93-111, 1988. · Zbl 0635.22003
[8] R. Jacoby, ”Some theorems on the structure of locally compact local groups,” Ann. of Math., vol. 66, pp. 36-69, 1957. · Zbl 0084.03202
[9] I. Kaplansky, Lie Algebras and Locally Compact Groups, Chicago, IL: The University of Chicago Press, 1971. · Zbl 0223.17001
[10] M. Kuranishi, ”On Euclidean local groups satisfying certain conditions,” Proc. Amer. Math. Soc., vol. 1, pp. 372-380, 1950. · Zbl 0038.01701
[11] T. McGaffey, A partial solution to a generalization of Hilbert’s local fifth problem: the standard part of a locally euclidean local nonstandard Lie group is an analytic Lie group.
[12] D. Montgomery and L. Zippin, Topological Transformation Groups, New York: Interscience Publishers, 1955. · Zbl 0068.01904
[13] D. Montgomery and L. Zippin, ”Small subgroups of finite-dimensional groups,” Ann. of Math., vol. 56, pp. 213-241, 1952. · Zbl 0049.30106
[14] P. J. Olver, ”Non-associative local Lie groups,” J. Lie Theory, vol. 6, iss. 1, pp. 23-51, 1996. · Zbl 0862.22005
[15] C. Plaut, ”Associativity and the local version of Hilbert’s fifth problem,” University of Tenessee, notes , 1993.
[16] L. Pontrjagin, Topological Group, Princeton, NJ: Princeton Univ. Press, 1939, vol. 2. · JFM 65.0872.02
[17] M. Singer, ”One parameter subgroups and nonstandard analysis,” Manuscripta Math., vol. 18, iss. 1, pp. 1-13, 1976. · Zbl 0328.22011
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