Banach-Stone theorems for Banach manifolds

Abstract

This short article addresses natural problems such as this one: Let M and N be two Banach manifolds such that the algebras of real-analytic functions on M and N are isomorphic as algebras. Does it follow that M and N are real-analytic isomorphic? The obvious way to attack the question is to identify, if possible, the sets M and N with the spectra of the relevant algebras, and then to transpose the algebra isomorphism. This often works, as shown in this article, but not always: an interesting example (Proposition 6) is given by M=c 0 (Γ) , where Γ is an uncountable set, and N=M∖{0} . This should be compared with P. Hajek's theorem [Israel J. Math. 104 (1998), 17–27; which asserts that there is no C 2 smooth function on the space c 0 (Γ) which vanishes in exactly one point.