The point in weak semiprojectivity and AANR compacta

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. ample shows a C ⁎ -algebra A can fail to be WSP while its unitization A ˜ is WSP. Another example shows WSP1 is not closed under direct sums.