Liftable normal elements

Von | September 22, 2020

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. [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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