2015 Fiscal Year Research-status Report
コンパクト型可換群の構造及びMarkov稠密性を実現する群位相の導入の研究
| Project/Area Number |
26400091
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| Research Institution | Ehime University |
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Principal Investigator |
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | direct products / direct sums / topological group / topological vector space / dual group / reflexive group / multiplier convergence / compact |
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| Outline of Annual Research Achievements |
We introduce three new notions for a subset A of an abelian topological group G: that of an absolutely Cauchy summable subset, of an absolutely summable subset and of a topologically independent subset. With these three notions at hand, we prove that: (1) an abelian topological group contains a direct product (direct sum) of kappa-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality kappa; (2) a topological vector space contains the direct sum of countably many real lines as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains the direct product of countably many lines as its subspace if and only if it has a multiplier convergent series of non-zero elements. We answer a question of Husek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.
We prove that the group G of all homomorphisms from the Baer-Specker group to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product of continuum many copies of the integers Z, this provides an example of a countable free closed non-reflexive subgroup of the direct product of continuum many copies of the integer group Z, thereby answering a problem of Galindo, Recorder-Nunez and Tkachenko.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
While working on main topics of this research project, we discovered a new technique that, when combined with our own methods from previous publications, lead to a solution of the thirty years old problem of Comfort-Protasov-Remus on the existence of minimally almost periodic topologies on abelian groups, as well as a problem of Gabriyelyan about the realization of von Neuman kernel by some Hausdorff group topology. These unexpected new developments got us temporarily sidetracked from the original problems.
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| Strategy for Future Research Activity |
We shall investigate the realization of the algebraic (Markov-Zariski closure) of a given countable subset in some pseudocompact group topology on this group. We shall also investigate the existence of Hausdorff group topologies on abelian group with a compactness-like property sandwiched between countable compactness and pseudocompactness.
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| Causes of Carryover |
I was not able to attend an international conference that I was planning to attend due to unexpected work during the period of the conference.
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| Expenditure Plan for Carryover Budget |
We plan to invite Professors M. J. Chasco (University of Navarra, Spain) and M. Tkachenko (UAM, Mexico) to Ehime University for a joint research in topological groups. The Incurring Amount is to be used for covering their lodging and travel expenses.
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Research Products
(8 results)
