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Is there a terminology for the following property of $C^*$ algebra $A$:

For every $C^*$ algebra $B$ and surjective $C^*$ morphism $\phi: B\to A$, every normal element $y\in A$ admits a normal element $b\in B$ with $\phi(b)=y$

I learned from $BDF$ theory that the Calkin algebra does not satisfy this property. What is an example of an infinite dimensional algebra with this property? What is a simple infinite dimensional algebra with this property?Does matrix algebra satisfy that? Is this property preserved by tensor product?(any tensor product you whish to choose),

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    $\begingroup$ I doubt very much that any C*-algebra other than the complexes or matrix rings over them (or maybe finite products of them) satisfies this condition (assuming all algebras are unital). For example, take $B$ to be the C* algebra generated by the right (or left) shift on $l^2(N)$ ($N$ is the set of positive integers), and let $A = C(T)$ ($T$ = circle); we have an onto map $B \to A$, and the canonical generator $z$ cannot be lifted to a normal (since otherwise the shifts would be unitaries). Perhaps this type of example can be generalized. $\endgroup$
    – David Handelman
    Commented May 4, 2020 at 15:22
  • $\begingroup$ @DavidHandelman in this example I think the index function is not identically 0(on the complement of essential spectrum). But is this property obvious for matrix algebra? $\endgroup$
    – Ali Taghavi
    Commented May 4, 2020 at 19:04
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    $\begingroup$ Lin showed that normal elements can be lifted from a quotient in at least one non-trivial situation. H.Lin, Approximation by normal elements with finite spectra in simple AF-algebras, J. Operator Theory, 31 (1994), 83-98. $\endgroup$
    – Douglas Somerset
    Commented May 7, 2020 at 18:35
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    $\begingroup$ Let $A$ be the algebra compact operators on separable Hilbert space. Then given a normal compact operator, one could map its spectrum homeomorphically into the real line, lift the resulting self-adjoint element to a self-adjoint element, and then undo the homeomorphism to get a normal lift of the original operator. So I think that the compacts give an infinite-dimensional simple example. I would suggest that $A$ must have the spectrum of every normal element homeomorphic to a subset of the reals. $\endgroup$
    – Douglas Somerset
    Commented May 7, 2020 at 20:49
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    $\begingroup$ Every projective C*-algebra has this property: A is projective if and only if for every surjective -homomorphism $\phi\colon B\to A$ there exists a *-homomorphism (a split, or section) $\sigma\colon A\to B$ such that $\phi\circ\sigma=\mathrm{id}_A$. If $y\in A$ is normal, then $\sigma(y)$ is a normal lift. There are infinite-dimensional projective C-algebras, for example any free product $\ast_I C_0((0,1])$. $\endgroup$
    – Hannes Thiel
    Commented May 11, 2020 at 12:41

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