The Point in Weak Semiprojectivity and AANR Compacta

TL;DR

This paper explores the properties of pointed approximative absolute neighborhood retracts and their implications for weak semiprojectivity in C*-algebras, revealing unexpected behaviors and distinctions between related properties.

Contribution

It introduces the concept of pointed approximative absolute neighborhood retracts and provides new examples illustrating differences in weak semiprojectivity properties of C*-algebras.

Findings

A C*-algebra can fail to be WSP while its unitization is WSP.

WSP1 is not closed under direct sums.

New examples of C*-algebras with unexpected weak semiprojectivity behavior.

Abstract

We initiate the study of pointed approximative absolute neighborhood retracts. Our motivation is to generate examples of C*-algebras that behave in unexpected ways with respect to weak semiprojectivity. We consider both weak semiprojectivity (WSP) and weak semiprojectivity with respect to the class of unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different properties. One example shows a C*-algebra can fail to be WSP while its unitization is WSP. Another example shows WSP1 is not closed under direct sums.