J. Aust. Math. Soc.  77 (2004), 17-28
Types over $C(K)$ spaces

Markus Pomper
  Indiana University East
  Richmond, IN

Let $K$ be a compact Hausdorff space and $C(K)$ the Banach space of all real-valued continuous functions on $K$, with the sup norm. Types over $C(K)$ (in the sense of Krivine and Maurey) are represented here by pairs $(l, u)$ of bounded real-valued functions on $K$, where $l$ is lower semicontinuous and $u$ is upper semicontinuous, $l \leq u$ and $l(x) = u(x)$ for every isolated point $x$ of $K$. For each pair the corresponding type is defined by the equation $\tau(g) = \max\{ \|l+g\|_\infty, \|u+g\|_\infty \}$ for all $g \in C(K)$, where $\| \cdot \|_\infty$ is the sup norm on bounded functions. The correspondence between types and pairs $(l,u)$ is bijective.
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