# A characterization of semiprojectivity for commutative C*‐algebras

@article{Srensen2011ACO,
title={A characterization of semiprojectivity for commutative C*‐algebras},
author={Adam P. W. S{\o}rensen and Hannes Thiel},
journal={Proceedings of the London Mathematical Society},
year={2011},
volume={105}
}
• Published 10 January 2011
• Mathematics
• Proceedings of the London Mathematical Society
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.
A characterization of weak (semi-)projectivity for commutative C*-algebras
We show that the spectrum X of a weakly semiprojective, commutative C*-algebra C(X) is at most one dimensional. This completes the work of S{\o}rensen and Thiel on the characterization of weak
The homotopy lifting theorem for semiprojective C*-algebras
We prove a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semiprojective C*-algebras. We also obtain some other results about semiprojective C*-algebras: a partial lifting
A characterization of semiprojectivity for subhomogeneous C*-algebras
We study semiprojective, subhomogeneous C*-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous
Obstructions to countable saturation in corona algebras
• Mathematics
• 2022
. We study the extent of countable saturation for coronas of abelian C ∗ -algebras. In particular, we show that the corona algebra of C 0 ( R n ) is countably saturated if and only if n = 1.
A Characterization of Semiprojectivity for Subhomogeneous
We study semiprojective, subhomogeneous C∗-algebras and give a detailed description of their structure. In particular, we find two characterizations of semiprojectivity for subhomogeneous
There are no noncommutative soft maps
It is shown that for a map $f \colon X \to Y$ of compact spaces the unital $\ast$-homomorphism $C(f) \colon C(Y) \to C(X)$ is projective in the category $\operatorname{Mor}({\mathcal C}^{1})$
Semiprojectivity and semiinjectivity in different categories
Projectivity and injectivity are fundamental notions in category theory. We consider natural weakenings termed semiprojectivity and semiinjectivity, and study these concepts in different categories.
Almost commuting matrices, cohomology, and dimension
• Mathematics
• 2019
We investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative $C^*$-algebras $C(X)$ are matricially semiprojective.

## References

SHOWING 1-10 OF 42 REFERENCES
Weakly Projective C*-Algebras
The noncommutative analog of an approximative absolute retract (AAR) is introduced, a weakly projective C*-algebra. This property sits between being residually finite dimensional and projectivity.
Semiprojectivity for certain purely infinite $C^*$-algebras
It is proved that classifiable simple separable nuclear purely infinite C*-algebras having finitely generated K-theory and torsion-free K 1 are semiprojective. This is accomplished by exhibiting
On semiprojectivity of C*-algebras of directed graphs
It is shown that if E is a countable, transitive directed graph with finitely many vertices, then C * (E) is semiprojective.
Noncommutative semialgebraic sets and associated lifting problems
• Mathematics
• 2009
We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a
Fundamental groups of one-dimensional spaces
• Mathematics
• 2013
Let X be a metrizable one-dimensional continuum. In the present paper we describe the fundamental group of X as a subgroup of its Cech homotopy group. In particular, the elements of the Cech homotopy
Operator Algebras: Theory of C*-Algebras and von Neumann Algebras
Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.
Inductive limits of projective $C$*-algebras
• Hannes Thiel
• Mathematics
Journal of Noncommutative Geometry
• 2020
We show that a separable C*-algebra is an inductive limits of projective C*-algebras if and only if it has trivial shape, that is, if it is shape equivalent to the zero C*-algebra. In particular,
Classifying Homotopy Types of One-Dimensional Peano Continua
CLASSIFYING HOMOTOPY TYPES OF ONE-DIMENSIONAL PEANO CONTINUA Mark Meilstrup Department of Mathematics Master of Science Determining the homotopy type of one-dimensional Peano continua has been an