Naslov
The centre-quotient property and weak centrality for C*-algebras

Autori
Gogić, Ilja

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Skup
The 48th Canadian Operator Symposium

Mjesto i datum
Toronto, Kanada, 25.05.2020. - 29.05.2020

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
C∗-algebra ; Centre-quotient property ; Weak centrality ; Commutator

Sažetak
Let A be a C*-algebra with centre Z(A). If I is a closed two-sided ideal of A, it is immediate that (Z(A)+I)/I⊆Z(A/I). A C*-algebra A is said to have the centre-quotient property (shortly, the CQ-property) if for any closed two-sided ideal I of A, (Z(A)+I)/I=Z(A/I). By a famous result of Vesterstrom from 1971, a unital C*- algebra A has the CQ-property if and only if it is weakly central, that is for any pair of maximal ideals M and N of A, M∩Z(A)=N∩Z(A) implies M=N. The most prominent examples of weakly central C*-algebras A are those satisfying the Dixmier property, that is for each x∈A the closure of the convex hull of the unitary orbit of x intersects Z(A). In particular, von Neumann algebras are weakly central. In this talk we study weak centrality, the CQ-property and several equivalent conditions for general C*-algebras that are not necessarily unital. We then investigate the failure of weak centrality in two different ways. Firstly, we show that every C∗-algebra A has a largest weakly central ideal J_wc(A), which can be readily determined in several examples. Secondly, we study the set VA of individual elements of A which prevent the weak centrality (or the CQ-property) of A. The set VA is contained in the complement of J_wc(A) and, in certain cases, is somewhat smaller than one might expect. In the course of this, we address a fundamental lifting problem that is closely linked to the CQ-property: for a fixed 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 is a joint work with Robert J. Archbold (University of Aberdeen).

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA