Inductive limits of semiprojective C*-algebras

TL;DR

This paper establishes that the class of C*-algebras formed as inductive limits of semiprojective C*-algebras is closed under shape domination, expanding the understanding of their structural properties and inclusiveness.

Contribution

It proves closure properties of inductive limits of semiprojective C*-algebras under shape domination, shape, and homotopy equivalence, showing this class is quite large.

Findings

The class is closed under shape domination.

Includes stabilized C*-algebras of continuous functions on spheres.

Contains the stable suspension of nuclear C*-algebras with UCT and torsion-free K0.

Abstract

We prove closure properties for the class of C*-algebras that are inductive limits of semiprojective C*-algebras. Most importantly, we show that this class is closed under shape domination, and so in particular under shape and homotopy equivalence. It follows that the considered class is quite large. It contains for instance the stable suspension of any nuclear C*-algebra satisfying the UCT and with torsion-free $K_0$-group. In particular, the stabilized C*-algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives.