A characterization of semiprojectivity for subhomogeneous $C^{\ast }$-algebras

Abstract

We study semiprojective, subhomogeneous $C^{\ast }$-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous mbox$C^{\ast }$-algebras: one in terms of their primitive ideal spaces and one by means of special direct limit structures over one-dimensional NCCW complexes. These results are obtained by working out several new permanence results for semiprojectivity, including a complete description of its behavior with respect to extensions by homogeneous mbox$C^{\ast }$-algebras.