The centre-quotient property and weak centrality for $C^{\ast }$-algebras

Abstract

We give a number of equivalent conditions (including weak centrality) for a general $C^{\ast }$-algebra to have the centre-quotient property. We show that every $C^{\ast }$-algebra $A$ has a largest weakly central ideal $J_{wc}(A)$. For an ideal $I$ of a unital $C^{\ast }$-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 $V_A$ of elements of an arbitrary $C^{\ast }$-algebra $A$ which prevent $A$ from having the centre-quotient property. The complement $\mathrm{CQ}(A):=A\setminus V_A$ always contains $Z(A)+J_{wc}(A)$ (where $Z(A)$ is the centre of $A$), with equality if and only if $A/J_{wc}(A)$ is abelian. Otherwise, $\mathrm{CQ}(A)$ fails spectacularly to be a $C^{\ast }$-subalgebra of $A$.

32 pages, to appear in Int. Math. Res. Not. IMRN