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Robert J. Archbold

Publications and source records attributed to Robert J. Archbold.

3 recordsLinked to original sources
arXiv2021

Local variants of the Dixmier property and weak centrality for C*-algebras

We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descriptions of more tractable sets of elements, where the corresponding averaging can be done arbitrarily close to the centre. This is achieved through several "spectral conditions", involving numerical ranges and tracial states.

math.OA↗
arXiv2020

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 $A$.

math.OA↗
arXiv2015

Norms of inner derivations for multiplier algebras of C*-algebras and group C*-algebras, II

The derivation constant $K(A)\geq \frac{1}{2}$ has been extensively studied for \emph{unital} non-commutative $C^*$-algebras. In this paper, we investigate properties of $K(M(A))$ where $M(A)$ is the multiplier algebra of a non-unital $C^*$-algebra $A$. A number of general results are obtained which are then applied to the group $C^*$-algebras $A=C^*(G_N)$ where $G_N$ is the motion group $\R^N\rtimes SO(N)$. Utilising the rich topological structure of the unitary dual $\widehat{G_N}$, it is shown that, for $N\geq3$, $$K(M(C^*(G_N)))= \frac{1}{2}\left\lceil \frac{N}{2}\right\rceil.$$

math.OA↗