Linear isometries between subspaces of continuous functions
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- by Jesús Araujo and Juan J. Font PDF
- Trans. Amer. Math. Soc. 349 (1997), 413-428 Request permission
Abstract:
We say that a linear subspace $A$ of $C_0 (X)$ is strongly separating if given any pair of distinct points $x_1, x_2$ of the locally compact space $X$, then there exists $f \in A$ such that $\left | f(x_1 ) \right | \neq \left | f(x_2 ) \right |$. In this paper we prove that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain singular subspaces of the Shilov boundaries of $B$ and $A$, sending the Choquet boundary of $B$ onto the Choquet boundary of $A$. We also provide an example which shows that the above result is no longer true if we do not assume $A$ to be strongly separating. Furthermore we obtain the following multiplicative representation of $T$: $(Tf)(y)=a(y)f(h(y))$ for all $y \in \partial B$ and all $f \in A$, where $a$ is a unimodular scalar-valued continuous function on $\partial B$. These results contain and extend some others by Amir and Arbel, Holsztyński, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.References
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Additional Information
- Jesús Araujo
- Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Facultad de Ciencias, Avda. de los Castros, s. n., E-39071 Santander, Spain
- Email: araujoj@ccaix3.unican.es
- Juan J. Font
- Affiliation: Departamento de Matemáticas, Universitat Jaume I, Campus Penyeta Roja, E-12071 Castellón, Spain
- Email: font@mat.uji.es
- Received by editor(s): October 16, 1995
- Additional Notes: Research of the first author was supported in part by the Spanish Dirección General de Investigación Científica y Técnica (DGICYT, PS90-100).
Research of the second author was supported in part by Fundació Caixa Castelló, (A-39-MA) - © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 349 (1997), 413-428
- MSC (1991): Primary 46E15; Secondary 46E25
- DOI: https://doi.org/10.1090/S0002-9947-97-01713-3
- MathSciNet review: 1373627