engleski

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 $J_{; ; ; ; ; ; ; ; wc}; ; ; ; ; ; ; ; (A)$. 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 $V_A$ of elements of an arbitrary $C^*$-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 $C^*$- subalgebra of $A$.

C*-algebra ; centre-quotient property ; weak centrality ; commutator

nije evidentirano