Given a C*-algebra and a closed, two-sided ideal , is the image of the normal elements in under the quotient map a closed subset of ?
Equivalently, if is a sequence of normal elements in that converge to , and if each admits a normal lift in , does admit a normal lift?
This question is raised in Example 6.5 in . It is equivalent to the question of whether the commutative C*-algebra of continuous functions on the disc vanishing at zero is -closed in the sense of Definition 6.1 in .
Of related interest is the class of normal elements that are universally liftable: We say that a normal element in a C*-algebra is universally liftable if for every C*-algebra and every surjective *-homomorphism there exists a normal element with . One can show that a normal element is universally liftable if and only if there exists a projective C*-algebra , a normal element and a *-homomorphism with . (A C*-algebra is projective if for every C*-algebra , every closed, two-sided ideal , and every *-homomorphism there exists a *-homomorphism such that , where is the quotient map.) Indeed, for the forward direct, one uses that every C*-algebra is the quotient of a projective C*-algebra, and for the converse direction one applies the definition of projectivity. In particular, every normal element in a projective C*-algebra is universally liftable.
Question: Given a C*-algebra , is the set of universally liftable normal elements in closed? Is it open (relative to the set of normal elements)?
Question: Is there a projective C*-algebra and a normal element such that for every C*-algebra , a normal element is universally liftable if and only if there exists a *-homomorphism with ?
- B. Blackadar, The Homotopy Lifting Theorem for Semiprojective -Algebras, MATH. SCAND. (2016) 291. https://doi.org/10.7146/math.scand.a-23691.