Abstract
We study bijections between algebras of smooth functions preserving certain parts of its structure. In particular, we show that multiplicative bijections are implemented by diffeomorphisms and they are automatically algebra isomorphisms. This confirms a conjecture by Mrčun and Šemrl.
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Cabello Sánchez, F., Homomorphisms on lattices of continuous functions, Positivity12 (2008), 341–362.
Cabello Sánchez, F., Cabello Sánchez, J., Ercan, Z. and Önal, S., Memorandum on multiplicative bijections and order, to appear in Semigroup Forum.
Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman, Harlow, 1993.
Fabian, M. and Zizler, V., An elementary approach to some questions in higher order smoothness in Banach spaces, Extracta Math.14 (1999), 295–327.
Garrido, M. I., Jaramillo, J. Á. and Prieto, Á., Banach–Stone theorems for Banach manifolds, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.)94 (2000), 525–538.
Grabowski, J., Isomorphisms of algebras of smooth functions revisited, Arch. Math. (Basel )85 (2005), 190–196.
Kaplansky, I., Lattices of continuous functions, Bull. Amer. Math. Soc.53 (1947), 617–623.
Kriegl, A. and Michor, P. W., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, Providence, RI, 1997.
Lochan, R. and Strauss, D., Lattice homomorphisms of spaces of continuous functions, J. London Math. Soc.25 (1982), 379–384.
Mrčun, J., On isomorphisms of algebras of smooth functions, Proc. Amer. Math. Soc.133 (2005), 3109–3113.
Mrčun, J. and Šemrl, P., Multiplicative bijections between algebras of differentiable functions, Ann. Acad. Sci. Fenn. Math.32 (2007), 471–480.
Shirota, T., A generalization of a theorem of I. Kaplansky, Osaka Math. J.4 (1952), 121–132.
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Research supported by DGICYT projects MTM2004-02635 and MTM2007-6994-C02-02.
JCS was supported in part by a grant of the UEx (programa propio–acción 2).
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Cabello Sánchez, F., Cabello Sánchez, J. Some preserver problems on algebras of smooth functions. Ark Mat 48, 289–300 (2010). https://doi.org/10.1007/s11512-009-0097-1
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DOI: https://doi.org/10.1007/s11512-009-0097-1