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

TL;DR

This paper explores conditions under which a unital C(X)-algebra is properly infinite based on its fibers, linking the problem to K_1-injectivity and reducing it to specific C([0,1])-algebras.

Contribution

It reformulates the problem of proper infiniteness in C(X)-algebras in terms of K_1-injectivity and provides partial solutions, reducing the general question to a specific case.

Findings

Proper infiniteness of C(X)-algebras can be characterized via K_1-injectivity.

The problem reduces to analyzing a specific C([0,1])-algebra with properly infinite fibers.

Partial answers are given to the question of proper infiniteness in this context.

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 if every unital properly infinite C*-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.