×

Real-linear surjective isometries between function spaces. (English) Zbl 1372.46021

Let \(X\) be a compact Hausdorff space and denote by \(C(X)\) the Banach space of all complex-valued continuous functions on \(X\) with the supremum norm. A complex-linear subspace \(A\) of \(C(X)\) is called a function space if \(A\) contains the constant functions and separates the points of \(X\). Recall that the Choquet boundary \(\text{Ch} (A)\) of a function space \(A \subset C(X)\) is the subset of \(X\) consisting of all points \(x\) such that the point-evaluation functional \(\delta_x: A \rightarrow \mathbb C\), defined by \(\delta_x(f) =f(x)\), is an extreme point of the dual unit ball of \(A\). If \(S: A \rightarrow B\) is a complex-linear surjective isometry between function spaces \(A\) of \(C(X)\) and \(B\) of \(C(Y)\), then \(S\) is a unimodular-weighted composition operator induced by a surjective homeomorphism between the Choquet boundaries of \(B\) and \(A\) by the Banach-Stone theorem and its variants.
When \(S\) is assumed to be real-linear but not necessarily complex-linear, there often exist continuous maps \(\alpha: \text{Ch} (B) \rightarrow \mathbb T = \{ z \in \mathbb C: |z|=1\}\) and \(\varepsilon: \text{Ch} (B) \rightarrow \{-1,1\}\) and a homeomorphism \(\varphi: \text{Ch} (B) \rightarrow \text{Ch} (A)\) so that \(S\) takes the following form, which the authors call the canonical form: \[ Sf(y) = \text{Re} (\alpha(y)f(\varphi(y))) + i\varepsilon(y) \text{Im} (\alpha(y)f(\varphi(y))), \quad f \in A, \;y \in \text{Ch} (B). \] There exist real-linear surjective isometries between function spaces which are not of the canonical form [H. Koshimizu et al., J. Math. Anal. Appl. 413, No. 1, 229–241 (2014; Zbl 1328.46018)] and the existence of a real-linear surjective isometry \(S: A \rightarrow B\) of non-canonical form forces the space \(\text{Ch} (A)\) to contain a homeomorphic copy of the unit circle \(\mathbb T\).
The authors show that, for every real-linear surjective isometry \(S:A \rightarrow B\) between function spaces \(A\) and \(B\), one can associate a homeomorphism \(\sigma : \mathbb T \times \text{Ch} (B) \rightarrow \mathbb T \times \text{Ch} (A)\). An examination of this homeomorphism \(\sigma\) leads to a topological condition on a compact Hausdorff space \(X\) such that every real-linear surjective isometry on a function space of \(C(X)\) has the canonical form. Furthermore, the authors give new examples of real-linear surjective isometries which do not take the canonical form.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46B04 Isometric theory of Banach spaces

Citations:

Zbl 1328.46018
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Araujo, J.; Font, J. J., Linear isometries between subspaces of continuous functions, Trans. Am. Math. Soc., 349, 413-428 (1997) · Zbl 0869.46014
[2] Cambern, M., Isometries of certain Banach algebras, Stud. Math., 25, 217-225 (1964-1965) · Zbl 0196.42702
[3] deLeeuw, K.; Rudin, W.; Wermer, J., The isometries of some function spaces, Proc. Am. Math. Soc., 11, 694-698 (1960) · Zbl 0097.09802
[4] Ellis, A. J., Real characterizations of function algebras amongst function spaces, Bull. Lond. Math. Soc., 22, 381-385 (1990) · Zbl 0713.46016
[5] Fleming, R.; Jamison, J., Isometries on Banach Spaces; Function Spaces, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 129 (2003), CRC Press: CRC Press Boca Raton · Zbl 1011.46001
[6] Fleming, R.; Jamison, J., Isometries on Banach Spaces, vol. 2; Vector-Valued Function Spaces, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., vol. 138 (2008), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton
[7] Hatori, O.; Miura, T., Real linear isometries between function algebras. II, Cent. Eur. J. Math., 11, 1838-1842 (2013) · Zbl 1294.46042
[8] Al-Halees, H.; Fleming, R. J., Extreme point methods and Banach-Stone theorems, J. Aust. Math. Soc., 75, 125-143 (2003) · Zbl 1036.46004
[9] Jamshidi, A.; Sady, F., Real-linear isometries between certain subspaces of continuous functions, Cent. Eur. J. Math., 11, 2034-2043 (2013) · Zbl 1294.46043
[10] Jarosz, K., Isometries in semisimple commutative Banach algebras, Proc. Am. Math. Soc., 94, 65-71 (1985) · Zbl 0577.46052
[11] Jarosz, K.; Pathak, V. D., Isometries between function spaces, Trans. Am. Math. Soc., 305, 193-206 (1988) · Zbl 0649.46024
[12] Jarosz, K.; Pathak, V. D., Isometries and small bound isomorphisms of function spaces, (Function Spaces. Function Spaces, Edwardsville, IL, 1990. Function Spaces. Function Spaces, Edwardsville, IL, 1990, Lect. Notes Pure Appl. Math., vol. 136 (1992), Marcel Dekker), 241-271 · Zbl 0804.46030
[13] Kawamura, K., Linear surjective isometries between vector-valued function spaces, J. Aust. Math. Soc., 100, 349-373 (2016) · Zbl 1352.46010
[14] Koshimizu, H.; Miura, T.; Takagi, H.; Takahasi, S.-E., Real-linear isometries between subspaces of continuous functions, J. Math. Anal. Appl., 413, 229-241 (2014) · Zbl 1328.46018
[15] Miura, T., Real-linear isometries between function algebras, Cent. Eur. J. Math., 9, 778-788 (2011) · Zbl 1243.46043
[16] Miura, T., Surjective isometries between function spaces, Contemp. Math., 645, 231-239 (2015) · Zbl 1352.46011
[17] Nagasawa, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Semin. Rep., 11, 182-188 (1959) · Zbl 0166.40002
[18] Novinger, W., Linear isometries of subspaces of spaces of continuous functions, Stud. Math., 53, 273-276 (1975) · Zbl 0273.46015
[19] Pathak, V. D., Isometries of \(C^{(n)} [0, 1]\), Pac. J. Math., 94, 211-222 (1981) · Zbl 0459.46037
[20] Väisälä, J., A proof of the Mazur-Ulam theorem, Am. Math. Mon., 110, 7, 633-635 (2003) · Zbl 1046.46017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.