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  • TYPE 2 SEMI-ALGEBRAS OF CONTINUOUS FUNCTIONS

  • F. F. BONSALL
  • Journal:
    Proceedings of the London Mathematical Society / Volume 81 / Issue 3 / November 2000
    Published online by Cambridge University Press:
    03 November 2000, pp. 725-746
    Print publication:
    November 2000

  • A semi-algebra of continuous functions is a cone$A$ of continuous real functions on a compactHausdorff space $X$ such that $A$ contains the products of its elements. A cone $A$ is said to be of type $n$ if $f\in A$ implies$f^n(1 + f)^{-1} \in A$. Uniformly closedsemi-algebras of types 0 and 1 have long beencharacterized in a manner analogous to the Stone--Weierstrass theorem, but, except forthe case when $A$ is generated by a singlefunction, little has been known about type 2.Here, progress is reported on two problems.The first is the characterization of those continuous linear functionals on $C(X)$ that determine semi-algebras of type 2. The secondis the determination of the type of the tensorproduct of two type 1 semi-algebras. 1991 Mathematics Subject Classification: 46J10.