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REFLEXIVITY INDEX AND IRRATIONAL ROTATIONS

Part of: Lattices Miscellaneous applications of functional analysis

Published online by Cambridge University Press:  29 March 2021

BINGZHANG MA Open the ORCID record for BINGZHANG MA [Opens in a new window]  and
K. J. HARRISON Open the ORCID record for K. J. HARRISON [Opens in a new window]
BINGZHANG MA*
Affiliation:
School of Science, East China University of Science and Technology, Shanghai, P. R. China
K. J. HARRISON
Affiliation:
School of Science, Murdoch University, Western Australia6150, Australia e-mail: K.Harrison@murdoch.edu.au
*
e-mail: bingzhangmath@foxmail.com
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Abstract

We determine the reflexivity index of some closed set lattices by constructing maps relative to irrational rotations. For example, various nests of closed balls and some topological spaces, such as even-dimensional spheres and a wedge of two circles, have reflexivity index 2. We also show that a connected double of spheres has reflexivity index at most 2.

Keywords

closed set latticeirrational rotationreflexivity index

MSC classification

Primary: 46N99: None of the above, but in this section
Secondary: 06B30: Topological lattices, order topologies
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 493 - 505
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

This research was partly supported by the National Natural Science Foundation of China (Grant No. 11871021).

References

Davidson, K. R., Nest Algebras: Triangular Forms for Operator Algebras on Hilbert Space (Longman, New York, 1988).Google Scholar
Halmos, P. R., ‘Reflexive lattices of subspaces’, J. Lond. Math. Soc. 4(2) (1971), 257–263.10.1112/jlms/s2-4.2.257CrossRefGoogle Scholar
Harrison, K. J. and Ward, J. A., ‘The reflexivity index of a lattice of sets’, J. Aust. Math. Soc. 97 (2014), 237–250.CrossRefGoogle Scholar
Harrison, K. J. and Ward, J. A., ‘Reflexive nests of finite subsets of a Banach space’, J. Math. Anal. Appl. 420 (2014), 1468–1477.10.1016/j.jmaa.2014.05.029CrossRefGoogle Scholar
Harrison, K. J. and Ward, J. A., ‘The reflexivity index of a finite distributive lattice of subspaces’, Linear Algebra Appl. 455 (2014), 73–81.CrossRefGoogle Scholar
Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, UK, 2002).Google Scholar
Kronecker, L., Näherungsweise ganzzahlige Auflösung linearer Gleichungen (Chelsea, New York, 1968).Google Scholar
Longstaff, W. E., ‘Reflexive index of a family of subspaces’, Bull. Aust. Math. Soc. 90 (2014), 134–140.CrossRefGoogle Scholar
Rotman, J. J., An Introduction to Algebraic Topology (Springer, New York, 1988).10.1007/978-1-4612-4576-6CrossRefGoogle Scholar
Zhao, D. S., ‘Reflexive index of a family of sets’, Kyungpook Math. J. 54 (2014), 263–269.CrossRefGoogle Scholar