Separation properties in the primitive ideal space of a multiplier algebra

2013

Proc. Amer. Math. Soc.

A condition on a derivation of an arbitrary C*-algebra is presented entailing that it is implemented as an inner derivation by a local multiplier. It is an outstanding open question whether every derivation of a C*-algebra A can be implemented as an inner derivation by a local multiplier, that is, an element in the direct limit of the multiplier algebras of the closed essential ideals of A. An affirmative answer was given by Elliott [4] for AF-algebras, and by Pedersen [11] for general separable C*-algebras. In fact, it suffices to assume that every closed essential ideal of A is σ-unital; hence Pedersen's result entails Sakai's theorem that every derivation of a simple unital C*-algebra is inner. But only an affirmative answer in the non-separable case would cover, extend and unify the results that every derivation of a simple C*-algebra is inner in the multiplier algebra [13] and that all derivations of von Neumann algebras [6], [12] and AW*-algebras [10] are inner. This quest becomes even more attractive by the recent results in [9] and [14] implying that, if a derivation δ on A is inner in the multiplier algebra, then there is a local multiplier a of A implementing δ such that δ = 2 a. No progress on the above question seems to have been made since it was raised in [11] (see also [4]). The purpose of this note is to present a criterion on a given derivation δ of a (possibly non-separable) C*-algebra A implying that δ is inner in the local multiplier algebra M loc (A). Though this criterion, inspired by Herstein's work [5], is rather algebraic in nature, it is hoped that some approximate version may eventually yield a positive solution of the general problem. 1. Notation and preliminaries Throughout this paper, M (A) will denote the multiplier algebra of the C*-algebra A. A left ideal L of A is said to be essential if its left annihilator L ⊥ = {a ∈ A | aL = 0} is zero. For a (closed) two-sided ideal I, the left annihilator coincides with the right annihilator, and I + I ⊥ is a (closed) essential ideal. Given two closed essential ideals I, J in A such that J ⊆ I, J is an essential ideal in M (I) and hence M (I) embeds isometrically into M (J). Forming the C*-direct limit of the directed family of multiplier algebras so obtained yields the local multiplier algebra of A,

Derivations of subhomogeneous $C^*$-algebras are implemented by local multipliers

2013

Proc. Amer. Math. Soc.

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

Archbold

International Mathematics Research Notices

2020

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 $A$.

The closure of two-sided multiplications on C*-algebras and phantom line bundles

Timoney

2016

Int Math Res Notices

Abstract. For a C * -algebra A we consider the problem of when the set TM 0 (A) of all two-sided multiplications x → axb (a, b ∈ A) on A is norm closed, as a subset of B(A). We first show that TM 0 (A) is norm closed for all prime C * -algebras A. On the other hand, if A ∼ = Γ 0 (E) is an n-homogeneous C * -algebra, where E is the canonical Mn-bundle over the primitive spectrum X of A, we show that TM 0 (A) fails to be norm closed if and only if there exists a σ-compact open subset U of X and a phantom complex line subbundle L of E over U (i.e. L is not globally trivial, but is trivial on all compact subsets of U ). This phenomenon occurs whenever n ≥ 2 and X is a CW-complex (or a topological manifold) of dimension 3 ≤ d < ∞.