Tokyo Journal of Mathematics

Peripheral Multiplicativity of Maps on Uniformly Closed Algebras of Continuous Functions Which Vanish at Infinity

Osamu HATORI, Takeshi MIURA, Hirokazu OKA, and Hiroyuki TAKAGI

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Abstract

We study maps between uniformly closed algebras of complex-valued continuous functions which vanish at infinity on locally compact Hausdorff spaces. Without assuming linearity nor multiplicativity on the maps we show that they are isometrical isomorphisms as Banach space operators if they satisfy that the peripheral range of the product of the images of any two elements coincides with the peripheral range of the product of those elements. Furthermore, if the underlying algebras contain approximate identities, then they are isometrically isomorphic as Banach algebras, which is a generalization of a recent result of Luttman and Tonev for the case of uniform algebras. On the other hand it is not the case without assuming the existence of approximate identities; An example is given.

Article information

Source
Tokyo J. Math., Volume 32, Number 1 (2009), 91-104.

Dates
First available in Project Euclid: 7 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1249648411

Digital Object Identifier
doi:10.3836/tjm/1249648411

Mathematical Reviews number (MathSciNet)
MR2541156

Zentralblatt MATH identifier
1201.46046

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]

Citation

HATORI, Osamu; MIURA, Takeshi; OKA, Hirokazu; TAKAGI, Hiroyuki. Peripheral Multiplicativity of Maps on Uniformly Closed Algebras of Continuous Functions Which Vanish at Infinity. Tokyo J. Math. 32 (2009), no. 1, 91--104. doi:10.3836/tjm/1249648411. https://projecteuclid.org/euclid.tjm/1249648411


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