A generalization of a theorem on biseparating maps
by
Melvin Henriksen
Harvey Mudd College

A \Phi-algebra is an Archimedean lattice-ordered algebra with an identity element that is a weak order unit. A bijection between lattice-ordered algebras A and B such that both T and T^-1 preserve orthogonality is said to be biseparating. In [J. Math. Ana. App., 12 (1995), 258-265], Araujo, Beckenstein and Narici proved that for any linear biseparating map T from C(X) onto C(Y), where X and Y are completely regular, there exist w in C(Y) and an homeomorphism h from the realcompactification \upsilonX of X onto \upsilonY, such that T(f)(y)=w(y).f(h(y)) for all f in C(X) and y in Y. In [Contemp. Math. 253, Amer. Math. Soc. 2000], M. Henriksen and F.A. Smith asked to what extent the result above can be generalized to a larger class of algebras and gave a partial answer. In the present paper, their answer to that question is generalized as follows. Let A and B be uniformly closed \Phi-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically such a weighted isomorphism, that is, there exist w in B and a lattice and algebra isomorphism S between A and B such that S(a)=wS(a) for all a in A. We then assume that every universally \sigma-complete projection band in A is essentially one dimensional. Under this extra condition and according to a result from the memoir [Memoir Amer. Math. Soc. 143 (2000), no 679] by Abramovich and Kitover, any linear biseparating map T from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, this later result is a generalization of the aforementioned theorem by Araujo et al. Since we also prove that every universally \sigma-complete projection band in the uniformly closed \Phi-algebra C(X) is essentially one-dimensional. The extent to which this result is best possible is discussed. This is part of joint research with K. Boulabiar and G. Buskes

Date received: May 19, 2002