On maximally q-positive sets

In his recent book From Hahn-Banach to monotonicity (Springer, Berlin, 2008), S. Simons has introduced the notion of SSD space to provide an abstract algebraic framework for the study of monotonicity. Graphs of (maximal) monotone operators appear to be (maximally) q-positive sets in suitably defined SSD spaces. The richer concept of SSDB space involves also a Banach space structure. In this paper we prove that the analog of the Fitzpatrick function of a maximally q-positive subset M in a SSD space (B, ⌊·,·, ⌋) is the smallest convex representation of M. As a consequence of this result it follows that, in the case of a SSDB space, the conjugate with respect to the pairing ⌊·,·,⌋ of any convex representation of M provides a convex representation of M, too. We also give a new proof of a characterization of maximally q-positive subsets of SSDB spaces in terms of such special representations. © Heldermann Verlag.