Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Isometries in semisimple, commutative Banach algebras

Author: Krzysztof Jarosz
Journal: Proc. Amer. Math. Soc. 94 (1985), 65-71
MSC: Primary 46J05
MathSciNet review: 781058
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that for any semisimple, commutative, complex Banach algebra $ A$ with unit there are norms on $ A$, which we call natural norms, equivalent to the original norm on $ A$ with the following property: Let $ (A,\vert\vert \cdot \vert{\vert _A},{e_A})$ and $ (B,\vert\vert \cdot \vert{\vert _B},{e_B})$ are commutative, semisimple Banach algebras with units and natural norms. Assume $ T$ is a linear isometry from $ (A,\vert\vert \cdot \vert{\vert _A})$ onto $ (B,\vert\vert \cdot \vert{\vert _B})$ with $ T{e_A} = {e_B}$. Then $ T$ is an isomorphism in the category of Banach algebras. For a fairly large class of algebras, for example, for uniform algebras, for algebras of the form $ {C^k}(X),{\text{ Lip}}(X),{\text{ AC}}(X)$, the natural norm we have defined coincides with a usual norm.

References [Enhancements On Off] (What's this?)

  • [1] M. Cambern, Isometries of certain Banach algebras, Studia Math. 25 (1965), 217-225. MR 0172129 (30:2355)
  • [2] M. Cambern and V. Pathak, Isometries of spaces of differentiable functions, Math. Japan. 26 (1981), 253-260. MR 624212 (82h:46043)
  • [3] T. W. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. MR 0410387 (53:14137)
  • [4] M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kōdai Math. Sem. Rep. 11 (1959), 182-188. MR 0121645 (22:12379)
  • [5] V. Pathak, Isometries of $ {C^{(n)}}[0,1]$, Pacific J. Math. 94 (1981), 211-222. MR 625820 (82i:46040)
  • [6] -, Linear isometries of spaces of absolutely continuous functions, Canad. J. Math. 34 (1982), 298-306. MR 658967 (83f:46023)
  • [7] N. V. Rao and A. K. Roy, Linear isometries of some function spaces, Pacific J. Math. 38 (1971), 177-192. MR 0308763 (46:7877)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46J05

Retrieve articles in all journals with MSC: 46J05

Additional Information

Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society