Abstract
Given a compact metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighbourhood retract of dimension at most 1. This confirms a conjecture of Blackadar.
Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As applications of our results, we prove two theorems about the structure of semiprojective commutative C*-algebras. Letting A be a commutative C*-algebra, we show firstly: If I is an ideal of A and A/I is finite-dimensional, then A is semiprojective if and only if I is; and secondly: A is semiprojective if and only if M2(A) is. This answers two questions about semiprojective C*-algebras in the commutative case.
Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As applications of our results, we prove two theorems about the structure of semiprojective commutative C*-algebras. Letting A be a commutative C*-algebra, we show firstly: If I is an ideal of A and A/I is finite-dimensional, then A is semiprojective if and only if I is; and secondly: A is semiprojective if and only if M2(A) is. This answers two questions about semiprojective C*-algebras in the commutative case.
| Original language | English |
|---|---|
| Journal | Proceedings of the London Mathematical Society |
| Volume | 105 |
| Issue number | 5 |
| Pages (from-to) | 1021-1046 |
| ISSN | 0024-6115 |
| DOIs | |
| Publication status | Published - Nov 2012 |