The Centre-Quotient Property and Weak Centrality forC*-Algebras

Abstract

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 $\textrm{CQ}(A):= A \setminus V_A$ 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, $\textrm{CQ}(A)$ fails spectacularly to be a $C^*$-subalgebra of $A$.

Which states can be reached from a given state by unital completely positive maps?

“…By Remark 4.3 a state on is maximally mixed if and only if it is maximally mixed on . By [3, 3.10], the quotients of weakly central C-algebras are weakly central, so in particular is weakly central. In this way, we reduce the proof to the algebra (instead of ), which has strong radical .…”

Local Variants of the Dixmier Property and Weak Centrality for C*-algebras

“…By a well-known result of Vesterstrøm [24] a unital C * -algebra A is weakly central if and only if it satisfies the centre-quotient property (the CQ-property), that is for any closed two-sided ideal I of A, (Z(A) + I)/I = Z(A/I). Motivated by results in [9], where the setting is purely algebraic, the first two authors defined in [5] a local version of the CQ-property as follows. An element a ∈ A is called a CQ-element if for every closed two-sided ideal I of A, a + I ∈ Z(A/I) implies a ∈ Z(A) + I ([5, Definition 4.1]).…”

“…An element a ∈ A is called a CQ-element if for every closed two-sided ideal I of A, a + I ∈ Z(A/I) implies a ∈ Z(A) + I ([5, Definition 4.1]). In [5,Theorem 4.8], a description of the set CQ(A) of all CQ-elements of A was obtained in terms of those maximal ideals of A which witness the failure of the weak centrality of A. Further, it was shown that CQ(A) contains all commutators and products by quasinilpotent elements.…”

“…However, from both the algebraic and topological viewpoint the set CQ(A) is in general rather strange. It always contains Z(A) + J wc (A) (where J wc (A) is the largest weakly central ideal of A, see [5,Theorem 3.22]), with equality if and only if A/J wc (A) is abelian. Otherwise it fails dramatically to be a C * -subalgebra of A: it is not norm-closed and it is neither closed under addition nor closed under multiplication.…”

“…In this paper we study local versions of the Dixmier property and of weak centrality, the latter motivated by Magajna's characterization in terms of EUCP maps. More precisely, given an element a ∈ A, we call a a As well as using earlier results on the Dixmier property and weak centrality [5,6,15,16,24], the methods of this paper rely on the Dauns-Hofmann theorem [22], Michael's selection theorem [18] and results on commutators, square zero elements and their lifts [1,17,20,21,23].…”