Deformations of infinite projections

Abstract

Let $A=(A_x)$ be a (semi-)continuous field of $C^{\ast }$-algebras over a compact Hausdorff space $X$ and let $p=(p_x)$ be a projection in $A$ such that each $p_x\in A_x$ is properly infinite ($x\in X$). Then $p$ is properly infinite if the field $A$ is upper semi-continuous. However, $p$ can be finite if the field is lower semi-continuous.