Measuring noncommutativity in C*-algebras

Robert J Archbold, Douglas W B Somerset

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

It is well known that if A is a von Neumann algebra then the norm of any inner derivation ad(a) is equal to twice the distance from the element a to the centre Z(A) of the algebra. More generally, this property holds in a unital C*-algebra if and only if the ideal P boolean AND Q boolean AND R is primal whenever P, Q, and R are primitive ideals of A such that P boolean AND Z(A) = Q boolean AND Z(A) = R boolean AND Z(A). In this paper we give a characterization, in terms of ideal structure, of those unital C*-algebras A for which the norm of any inner derivation ad(a) at least dominates the distance from a to the centre Z(A). In doing so, we show that if A does not have this property then it necessarily contains an element a, with parallel to ad(a)parallel to = 1, whose distance from Z(A) is greater than or equal to 3+8 root 2/14.. We also show how this number is related to the numbers 4/root 15 and 1/2 + 1/root 3 which have previously arisen in the study of norms of inner derivations. The techniques used in this work include spectral theory, the theory of primitive and primal ideals, and constrained geometrical optimisation. (C) 2006 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)247-271
Number of pages25
JournalJournal of Functional Analysis
Volume242
Issue number1
Early online date14 Aug 2006
DOIs
Publication statusPublished - 1 Jan 2007

Keywords

  • C*-algebra
  • inner derivation
  • primitive ideal
  • primal ideal
  • spectral theory
  • geometric optimisation
  • inner derivations
  • primal ideals
  • norm

Cite this