Miao, Xinhe; Cao, Jiling; Xiong, Hongyun Banach-Stone theorems and Riesz algebras. (English) Zbl 1102.46017 J. Math. Anal. Appl. 313, No. 1, 177-183 (2006). Let \(X, Y\) be compact Hausdorff spaces and let \(E, F\) be both Banach lattices and Riesz algebras. In this paper, the following main result shall be proved: If \(F\) has no zero-divisor and there exists a Riesz algebraic isomorphism \(\Phi:C(X,E)\to C(Y,F)\) such that \(\Phi(f)\) has no zero if \(f\) has none, then \(X\) is homeomorphic to \(Y\) and \(E\) is Riesz algebraically isomorphic to \(F\). Cited in 1 ReviewCited in 7 Documents MSC: 46B42 Banach lattices 46H05 General theory of topological algebras Keywords:Banach lattice; Riesz algebra; Riesz algebraic isomorphism; Support; Banach-Stone theorem PDFBibTeX XMLCite \textit{X. Miao} et al., J. Math. Anal. Appl. 313, No. 1, 177--183 (2006; Zbl 1102.46017) Full Text: DOI References: [1] Araujo, J.; Beckenstein, E.; Narici, L., Biseparating maps and homeomorphic real-compactifications, J. Math. Anal. Appl., 192, 258-265 (1995) · Zbl 0828.47024 [2] Cao, J.; Reilly, I.; Xiong, H., A lattice-valued Banach-Stone theorem, Acta Math. Hungar., 98, 103-110 (2003) · Zbl 1027.46025 [3] de Jonge, E.; van Rooij, A., Introduction to Riesz Space, Math. Centre Tracts, vol. 78 (1978), Mathematisch Centrum: Mathematisch Centrum Amsterdam [4] Gillman, L.; Jerison, M., Rings of Continuous Functions (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0151.30003 [5] Hernandez, S.; Beckenstein, E.; Narici, L., Banach-Stone theorem and separating maps, Manuscripta Math., 86, 409-416 (1995) · Zbl 0827.46032 [6] Huijsmans, C. B.; de Pagter, B., Subalgebras and Riesz subspaces of an \(f\)-algebra, Proc. London Math. Soc., 48, 161-174 (1984) · Zbl 0534.46010 [7] Luxemberg, W.; Zaanen, A., Riesz Spaces I (1971), North-Holland: North-Holland Amsterdam [8] Meyer-Nieberg, P., Banach Lattices (1991), Springer-Verlag: Springer-Verlag Berlin · Zbl 0743.46015 [9] Xiong, H., A characterization of Riesz spaces which are Riesz isomorphic to \(C(X)\) for some completely regular space \(X\), Nederl. Akad. Wetensch. Indag. Math., 51, 87-95 (1989) · Zbl 0693.46003 [10] Zaanen, A. C., Riesz Spaces II (1983), North-Holland: North-Holland Amsterdam · Zbl 0519.46001 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.