Almost-Periodic Algebras and Their Automorphisms
Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 657-672.

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem concerning the form of the maximal ideal space of an almost-periodic algebra formed by functions on $\mathbb{R}^m$ is considered. It is shown that this space is homeomorphic to the topological group dual to the group of frequencies of the algebra under consideration. In the case of a quasiperiodic algebra, the mappings of $\mathbb{R}^n$ generating automorphisms of the algebra are described. Several specific examples are given and a relation to the theory of quasicrystals is indicated.
Keywords: maximal ideal space, almost-periodic algebra, dual group, automorphism, quasicrystal.
@article{MZM_2017_102_5_a2,
     author = {A. B. Antonevich and A. N. Buzulutskaya (Glaz)},
     title = {Almost-Periodic {Algebras} and {Their} {Automorphisms}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {657--672},
     publisher = {mathdoc},
     volume = {102},
     number = {5},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_5_a2/}
}
TY  - JOUR
AU  - A. B. Antonevich
AU  - A. N. Buzulutskaya (Glaz)
TI  - Almost-Periodic Algebras and Their Automorphisms
JO  - Matematičeskie zametki
PY  - 2017
SP  - 657
EP  - 672
VL  - 102
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2017_102_5_a2/
LA  - ru
ID  - MZM_2017_102_5_a2
ER  - 
%0 Journal Article
%A A. B. Antonevich
%A A. N. Buzulutskaya (Glaz)
%T Almost-Periodic Algebras and Their Automorphisms
%J Matematičeskie zametki
%D 2017
%P 657-672
%V 102
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2017_102_5_a2/
%G ru
%F MZM_2017_102_5_a2
A. B. Antonevich; A. N. Buzulutskaya (Glaz). Almost-Periodic Algebras and Their Automorphisms. Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 657-672. http://geodesic.mathdoc.fr/item/MZM_2017_102_5_a2/

[1] G. Bor, Pochti periodicheskie funktsii, URSS, M., 2009 | MR | Zbl

[2] B. M. Levitan, V. V. Zhikov, Pochti-periodicheskie funktsii i differentsialnye uravneniya, Izd-vo Mosk. un-ta, M., 1978 | MR | Zbl

[3] L. S. Pontryagin, Nepreryvnye gruppy, Nauka, M., 1973 | MR | Zbl

[4] W. Rudin, Fourier Analysis on Groups, Intersience Publ., New York, 1962 | MR | Zbl

[5] A. B. Katok, B. Khasselblat, Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Faktorial, M., 1999 | MR | Zbl

[6] I. A. Dynnikov, S. P. Novikov, “Topologiya kvaziperiodicheskikh funktsii na ploskosti”, UMN, 60:1 (361) (2005), 3–28 | DOI | MR | Zbl

[7] M. I. Rabinovich, A. L. Fabrikant, L. Sh. Tsimring, “Konechnomernyi prostranstvennyi besporyadok”, UFN, 162:8 (1992), 1–48 | DOI

[8] A. B. Antonevich, Lineinye funktsionalnye uravneniya. Operatornyi podkhod, Izd-vo “Universitetskoe”, Minsk, 1988 | MR | Zbl

[9] A. Antonevich, A. Lebedev, Functional-Differential Equations. I. $C^*$-theory, Pitman Monogr. Surveys Pure Appl. Math., 70, Longman Sci. and Tech., Harlow, 1994 | MR | Zbl

[10] G. J. Murphy, “Crossed product of $C^*$-algebras by an endomorphisms”, Integral Equation Operator Theory, 24:3 (1996), 298–319 | DOI | MR | Zbl

[11] R. Exel, “A new look at the crossed product of a $C^*$-algebra by an endomorphism”, Ergodic Theory Dynam. Systems, 23:6 (2003), 1733–1750 | DOI | MR | Zbl

[12] A. B. Antonevich, V. I. Bakhtin, A. V. Lebedev, “Skreschennoe proizvedenie $C^*$-algebry na endomorfizm, algebry koeffitsientov i transfer-operatory”, Matem. sb., 202:9 (2011), 3–34 | DOI | MR | Zbl

[13] B. Kvasnevski, A. V. Lebedev, “Obratimye rasshireniya neobratimykh dinamicheskikh sistem: $C^*$-metod”, Matem. sb., 199:11 (2008), 45–74 | DOI | MR | Zbl

[14] D. Shechtman, L. Blech, D. Gratias, J. W. Cahn, “Mettalic phase with long-range orientational order and no translational symmetry”, Phys. Rev. Lett., 53 (1984), 1951–1954 | DOI

[15] V. I. Arnold, Gyuigens i Barrou, Nyuton i Guk. Pervye shagi metematicheskogo analiza i teorii katastrof, ot evolvent do kvazikristallov, Nauka, M., 1989 | MR | Zbl

[16] Le Tkhang Ty Kuok, S. A. Piunikhin, V. A. Sadov, “Geometriya kvazikristallov”, UMN, 48:1 (289) (1993), 41–102 | MR | Zbl

[17] A. B. Antonevich, A. N. Glaz, “Kvaziperiodicheskie algebry, invariantnye otnositelnogo lineinogo otobrazheniya”, Dokl. NAN Belarusi, 5 (2014), 30–35