Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.