Tokyo Journal of Mathematics

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

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

https://projecteuclid.org/euclid.tjm/1249648411

Digital Object Identifier
doi:10.3836/tjm/1249648411

Mathematical Reviews number (MathSciNet)
MR2541156

Zentralblatt MATH identifier
1201.46046

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

References

• A. Browder, Introduction to Function Algebras, W.A. Benjamin, 1969.
• T. W. Gamelin, Uniform Algebras 2nd ed., Chelsea Publishing Company, 1984.
• O. Hatori, T. Miura and H. Oka, An example of multiplicatively spectrum-preserving maps between non-isomorphic semi-simple commutative Banach algebras, Nihonkai Math. J., 18 (2007), 11–15.
• O. Hatori, T. Miura and H. Takagi, Characterizations of isometric isomorphisms between uniform algebras via non-linear range-preserving properties, Proc. Amer. Math. Soc., 134 (2006), 2923–2930.
• O. Hatori, T. Miura and H. Takagi, Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl., 326 (2007), 281–296.
• D. Honma, Surjections on the algebras of continuous functions which preserve peripheral spectrum, Contemp. Math., 435 (2007), 199–205.
• A. Luttman and T. Tonev, Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 135 (2007), 3589–3598.
• L. Molnár, Some characterizations of the automorphisms of $B(H)$ and $C(X)$, Proc. Amer. Math. Soc., 130 (2002), 111–120.
• N. V. Rao and A. K. Roy, Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 133 (2005), 1135–1142.
• N. V. Rao and A. K. Roy, Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc., 48 (2005) 219–229.
• E. L. Stout, The Theory of Uniform Algebras, Bogden & Quigley, Inc. Publishers, 1971.