A note on David Harrison’s theory of preprimes

A Kadison–Dubois representation for associative rings

“…By going to the tensor product A ⊗ Z Q, we are immediately reduced to the case where A contains Q. For details, see [7] or [16]. Notation.…”

“…Assume (iii) holds. Then Arch(M ) satisÿes the KDP by the classical KadisonDubois representation given in [7]. Clearly, M ⊆ Arch(M ) implies X (Arch(M )) ⊆ X (M ).…”

“…We will show that [x; y] ∈ I (M ). As T is a preordering, M is an archimedean preprime and thus satisÿes the KDP by the original Kadison-Dubois representation [4,7]. But this means that A=I (M ) is a commutative ring (since it is a subring of the ring of all real-valued functions on a compact space), hence [x; y] ∈ I (M ).…”

“…The Kadison-Dubois representation has a remarkable history, starting from functional analytic proofs by Kadison [12] and Dubois [7]. Later algebraic proofs were discovered by Krivine [13,14] and Becker and Schwartz [3] (for commutative associative rings) and CimpriÄ c [4] (for noncommutative associative rings) giving representation spaces useful for applications in real algebraic geometry.…”

Duality and stability in extremum problems involving convex functions

“…We shall then speak of E and E* as topologically paired spaces. The theory of such pairings may be found in [1,Chap. IV].…”