Properly infinite $C(X)$-algebras and $K_1$-injectivity

Abstract

We investigate if a unital $C(X)$-algebra is properly infinite when all its fibres are properly infinite. We show that this question can be rephrased in several different ways, including the question of whether every unital properly infinite ${\mathrm{C}}^{\ast }$-algebra is $K_1$-injective. We provide partial answers to these questions, and we show that the general question on proper infiniteness of $C(X)$-algebras can be reduced to establishing proper infiniteness of a specific $C([0,1])$-algebra with properly infinite fibres.