Topological groups without infinite precompact continuous homomorphic images
Topological groups without infinite precompact continuous homomorphic images
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没有无限预紧连续同态图像的拓扑群
DOI：
10.1016/j.topol.2020.107544
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发表时间：
2020
影响因子：
0.6
通讯作者：
Yanez Victor Hugo
中科院分区：
文献类型：
--
作者：
吉村綾馬;金子正弘;梶原智之;小町守;Yanez Victor Hugo
设P是拓扑群的一个性质。我们称一个拓扑群G是Minap模P，如果它的像f[G]在从G到紧群K的每个连续同态f：G→K下具有性质P。当P是平凡群时，Minap模P群正是von Neumann和Wigner的极小概周期群。我们给出了这些新的群的五个性质P的特征：有限的、有界的、扭转的、紧的和连通的。我们证明了这五类是不同的，不同于极小概周期群的标准类。我们给每个Abelian群配备了Hausdorff MINAP模有限群拓扑，从而表明，与经典的极小几乎周期性不同，MINAP模有限性质对基础群的代数结构没有任何限制。最后，我们证明了每个拓扑群G关于其von Neumann核n(G)的商群G/n(G)是G在极大概周期(MAP)群类中的范畴映射，并且自然商映射起着反射同态的作用。因此，这个具有Bohr拓扑的商群(G/n(G))+在其Bohr紧化的典范同态下与G的象拓扑同构。
Let P be a property of topological groups. We say that a topological group G is MinAP modulo P if its image f [G] under each continuous homomorphism f: G→ K from G to a compact group K has property P. When P is the property of being the trivial group, MinAP modulo P groups are precisely the minimally almost periodic groups of von Neumann and Wigner. We give a characterization of these new classes of groups for five properties P: finite, bounded, torsion, compact and connected. We show that these five classes are distinct and differ from the standard class of minimally almost periodic groups. We equip every Abelian group with a Hausdorff MinAP modulo finite group topology, thereby showing that, unlike the classical minimal almost periodicity, the property MinAP modulo finite imposes no restrictions whatsoever on the algebraic structure of the underlying group. At last but not least, we prove that the quotient group G/n (G) of every topological group G with respect to its von Neumann kernel n (G) is the categorical reflection of G in the class of maximally almost periodic (MAP) groups, and the natural quotient map plays the role of reflection homomorphism. As a consequence, this quotient group (G/n (G))+ equipped with its Bohr topology is topologically isomorphic to the image of G under the canonical homomorphism to its Bohr compactification.