Synnatzschke's theorem for polynomials (Q830242)

From the authors' abstract: ``We establish a multilinear generalisation of Synnatzschke's Theorem [\textit{J.~Synnatzschke}, Vestn. Leningr. Univ., Mat. Mekh. Astron. 1972, No.~1, 60--69 (1972; Zbl 0234.47035)] for regular operators on Banach lattices, with the Arens adjoint taking the place of the transpose.'' Recall that a linear operator \(T:E \to F\), between Banach lattices with \(F\) Dedekind-complete, is said to be {\em regular} if \(T\) is the difference of two positive operators. Synnatzschke's theorem states that if \(T:E \to F\) is a regular operator, then the transpose \(T':F' \to E'\) is order continuous and its restriction to the order continuous dual of \(F$, $F'_n\), satisfies \(|T'| = |T|'\). Here, the focus is on regular multilinear maps \(A:E_1 \times \cdots \times E_m \to G\) from Banach lattices \(E_j\) to a Dedekind complete Banach lattice \(G\). (A {\em regular} multilinear mapping is defined to be a difference of two positive multilinear maps.) The authors adopt the method of \textit{R. Arens} [Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)] to construct the extension of \(A\) to \(A^{*(m+1)}\) on \((E'_1)'_n \times \cdots \times (E'_m)'_n\). Their main result is the following analogy of Synnatzschke's theorem: \(|A^{*(m+1)}| = |A|^{*(m+1)}\). Among a number of interesting consequences is a version of the Davie-Gamelin theorem for polynomials [\textit{A.~M. Davie} and \textit{T.~W. Gamelin}, Proc. Am. Math. Soc. 106, No.~2, 351--356 (1989; Zbl 0683.46037)].