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