Liftable normal elements

Given a C*-algebra $A$ and a closed, two-sided ideal $I\subseteq A$ , is the image of the normal elements in $A$ under the quotient map $A\to A/I$ a closed subset of $A/I$ ?

Equivalently, if $(x_n)_n$ is a sequence of normal elements in $A/I$ that converge to $x$ , and if each $x_n$ admits a normal lift in $A$ , does $x$ admit a normal lift?

This question is raised in Example 6.5 in [1]. It is equivalent to the question of whether the commutative C*-algebra of continuous functions on the disc vanishing at zero is $\ell$ -closed in the sense of Definition 6.1 in [1].

Of related interest is the class of normal elements that are universally liftable: We say that a normal element $a$ in a C*-algebra $A$ is universally liftable if for every C*-algebra $B$ and every surjective *-homomorphism $\pi\colon B\to A$ there exists a normal element $b\in B$ with $\pi(b)=a$ . One can show that a normal element $a\in A$ is universally liftable if and only if there exists a projective C*-algebra $P$ , a normal element $b\in P$ and a *-homomorphism $\varphi\colon P\to A$ with $\varphi(b)=a$ . (A C*-algebra $P$ is projective if for every C*-algebra $D$ , every closed, two-sided ideal $I\subseteq D$ , and every *-homomorphism $\psi\colon P\to D/I$ there exists a *-homomorphism $\tilde{\psi}\colon P\to D$ such that $\pi\circ\tilde{\psi}=\psi$ , where $\pi\colon D\to D/I$ 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 $A$ , is the set of universally liftable normal elements in $A$ closed? Is it open (relative to the set of normal elements)?

Question: Is there a projective C*-algebra $P$ and a normal element $b\in P$ such that for every C*-algebra $A$ , a normal element $a\in A$ is universally liftable if and only if there exists a *-homomorphism $\varphi\colon P\to A$ with $\varphi(b)=a$ ?

[1]
B. Blackadar, The Homotopy Lifting Theorem for Semiprojective $C^*$ -Algebras, MATH. SCAND. (2016) 291. https://doi.org/10.7146/math.scand.a-23691.

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