Synnatzschke’s theorem for polynomials

Abstract

We establish a multilinear generalisation of Synnatzschke’s Theorem for regular operators on Banach lattices, with the Arens adjoint taking the place of the transpose. Using this result, we show that the Aron–Berner extension of a regular homogeneous polynomial to the order continuous bidual preserves the absolute value. As a consequence, it follows that the regular norm is preserved by the Aron–Berner extension.