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Isomorphism of uniform algebras on the 2-torus

Published online by Cambridge University Press:  16 May 2018

JUSTIN R. PETERS  and
PREECHAYA SANYATIT
JUSTIN R. PETERS
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa, IA 50011U.S.A. e-mail: [email protected]
PREECHAYA SANYATIT
Affiliation:
Department of Mathematics, Silpakorn University, Nakhon Pathom, 73000Thailand e-mail: [email protected]
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Abstract

For α a positive irrational, let 𝛢α be the subalgebra of continuous functions on the two-torus whose Fourier transform vanishes at (m, n) if m+αn < 0. These algebras were studied by Wermer and others, who proved properties such as maximality and characterised the Gelfand space. One of the major themes of current work in operator algebras is classification, but none of the properties which were investigated earlier distinguished between 𝛢α and αβ, if β is another positive irrational. We address this question. We also determine the automorphism group of 𝛢α.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 89 - 106
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Elliott, G. A. and Niu, Z. On the classification of simple amenable C*-algebras with finite decomposition rank. Operator Algebras and Their Applications: A Tribute to Richard V. Kadison. Contemp. Math. 671 (2016), 117–125.Google Scholar
[2] Elliott, G. A., Gong, G., Lin, H. and Niu, Z.. On the classification of simple amenable C*-algebras with finite decomposition rank, II. arXiv:1507.03437.Google Scholar
[3] Tikuisis, A., White, S. and Winter, W. Quasidiagonality of nuclear C*-algebras, Ann. of Math. (2) 185, no. 1, 229–284, 2017.Google Scholar
[4] Douglas, R. Banach Algebra techniques in operator theory. 2nd edition, Graduate Texts in Math. 179 (Springer, 1998).Google Scholar
[5] Encyclopedia of Mathematics, www.encyclopediaofmath.org. Commutative Banach algebra.Google Scholar
[6] Gamelin, T. W. Uniform Algebras (Prentice–Hall, 1969).Google Scholar
[7] Hoffman, K. Banach Spaces of Analytic Functions (Prentice–Hall, 1962).Google Scholar
[8] Murphy, G. C*-Algebras and Operator Theory (Academic Press, 1990).Google Scholar
[9] Wermer, J. Banach Algebras and Analytic Functions. Adv. Math. 1 (1) (1961), 51–102.Google Scholar
[10] Stout, E. L. The Theory of Uniform Algebras (Bogden and Quigley, 1971).Google Scholar
[11] Barbeau, E. J. Pell's Equation (Springer–Verlag, 2002).Google Scholar
[12] Robertson, J. P. Solving the generalised Pell equation x2 - Dy2 = N. www.jpr2718.org/pell.pdf (July 2004).Google Scholar
[13] Arzhantseva, G., Lafont, J.-F. and Minasyan, A.. Isomorphism versus commensurability for a class of finitely presented groups. J. Group Theory 17 (2014), 361–378.Google Scholar
[14] Karpenkov, O. Geometry of Continued Fractions (Springer–Verlag, 2013).Google Scholar
[15] Zygmund, A. Trigonometric Series Vol II (Cambridge University Press, 1977).Google Scholar
[16] Sanyatit, P. Isomorphism of uniform algebras on the 2-torus. Graduate Theses and Dissertations. 16008. https://lib.dr.iastate.edu/etd/16008.Google Scholar