The centre-quotient property and weak centrality for C^∗-algebras

We give a number of equivalent conditions (including weak centrality) for a general C∗-algebra to have the centre-quotient property. We show that every C∗-algebra A has a largest weakly central ideal Jwc(A). For an ideal I of a unital C∗-algebra A, we find a necessary and sufficient condition for a central element of A/I to lift to a central element of A. This leads to a characterisation of the set VA of elements of an arbitrary C∗-algebra A which prevent A from having the centre-quotient property. The complement CQ(A) := A \ VA always contains Z(A) + Jwc(A) (where Z(A) is the centre of A), with equality if and only if A/Jwc(A) is abelian. Otherwise, CQ(A) fails spectacularly to be a C∗-subalgebra of A.