Geometry of a crossed product

Abstract

We introduce a continuous dimension function α : ∙ → R \alpha : \bullet \to \mathbb {R} on the Grothendieck group K 0 K_0 over the crossed product C ∗ C^* -algebra C ( X ) ⋊ ϕ Z C(X)\rtimes _{\phi }\mathbb {Z} . The function α \alpha has an elegant geometry: on every minimal flow ϕ t \phi ^t it takes the value of the “rotation number" of ϕ t \phi ^t ; such a problem was posed in 1936 by A. Weil.