Disjointness preserving shifts on C0(X)

Abstract

We study disjointness preserving (quasi-)n-shift operators on C 0 (X), where X is locally compact and Hausdorff. When C 0 (X) admits a quasi-n-shift T , there is a countable subset of X ∞ = X ∪ {∞} equipped with a tree-like structure, called ϕ-tree, with exactly n joints such that the action of T on C 0 (X) can be implemented as a shift on the ϕ-tree. If T is an n-shift, then the ϕ-tree is dense in X and thus X is separable. By analyzing the structure of the ϕ-tree, we show that every (quasi-)n-shift on c 0 can always be written as a product of n (quasi-)1-shifts. Although it is not the case for general C 0 (X) as shown by our counter examples, we can do so after dilation.