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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Linear isometries between subspaces
of continuous functions

Authors: Jesús Araujo and Juan J. Font
Journal: Trans. Amer. Math. Soc. 349 (1997), 413-428
MSC (1991): Primary 46E15; Secondary 46E25
MathSciNet review: 1373627
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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, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

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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

Juan J. Font
Affiliation: Departamento de Matemáticas, Universitat Jaume I, Campus Penyeta Roja, E-12071 Castellón, Spain

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)
Article copyright: © Copyright 1997 American Mathematical Society