Type 2 semi-algebras of continuous functions. (Q2766380)

Let \(X\) be a compact Hausdorff space and let \(C(X)\) be the Banach algebra of all real-valued continuous functions on \(X\). A cone in \(C(X)\) is a non-void subset \(A\) such that NEWLINE\[NEWLINE f,g\in A, \alpha \in \mathbb{R}^+ \Longrightarrow f+g, \alpha f\in A NEWLINE\]NEWLINE and NEWLINE\[NEWLINE f,-f\in A \Longrightarrow f=0. NEWLINE\]NEWLINE A semi-algebra is a cone \(A\) such that also NEWLINE\[NEWLINE f,g\in A \Longrightarrow fg\in A. NEWLINE\]NEWLINE A cone (or semi-algebra) \(A\) is of type 0 if NEWLINE\[NEWLINE f\in A \Longrightarrow (1+f)^{-1} \in A, NEWLINE\]NEWLINE and, for \(n\in \mathbb{N}\), \(A\) is of type \(n\) if NEWLINE\[NEWLINE f\in A \Longrightarrow f^n(1+f)^{-1} \in A. NEWLINE\]NEWLINE Complete characterizations of closed unital type 0 and type \(1\) semi-algebras were found by the author more than 40 years ago. Unfortunately, similar general results are not known even for type 2 semi-algebras. In this paper, after the introduction the author presents a finer analysis of type 1 semi-algebras which are used to obtain new contributions to the study of type 2 cones. Namely, some results are presented concerning the cone NEWLINE\[NEWLINE B(A,\phi)=\{f\in A\, :\, \phi(f)\geq 0\}, NEWLINE\]NEWLINE where \(A\) is a closed unital type 1 semi-algebra in \(C(X)\) which separates the points of \(X\) and \(\phi\) is a continuous linear functional on \(C(X)\). A widely applicable sufficient condition is obtained for \(B(A,\phi)\) to be a type 2 semi-algebra. Given a finitely supported functional \(\phi\), a necessary and sufficient condition is presented for \(B(A,\phi)\) to be a type 2 cone. In the last section, the author investigates the type of the tensor product of closed unital semi-algebras \(A\) and \(B\). The results can be summarized by saying that for certain classes of the underlying topological spaces, we have NEWLINE\[NEWLINE \text{{type}}\,(A\otimes B)=\text{{type}}\,A + \text{{type}}\,B NEWLINE\]NEWLINE provided that \(A\),\(B\) are of type 1 at most.