### 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 language | English |
---|---|

Pages (from-to) | 247-271 |

Number of pages | 25 |

Journal | Journal of Functional Analysis |

Volume | 242 |

Issue number | 1 |

Early online date | 14 Aug 2006 |

DOIs | |

Publication status | Published - 1 Jan 2007 |

### Keywords

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

### Cite this

*Journal of Functional Analysis*,

*242*(1), 247-271. https://doi.org/10.1016/j.jfa.2006.06.012

**Measuring noncommutativity in C*-algebras.** / Archbold, Robert J; Somerset, Douglas W B.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 242, no. 1, pp. 247-271. https://doi.org/10.1016/j.jfa.2006.06.012

}

TY - JOUR

T1 - Measuring noncommutativity in C*-algebras

AU - Archbold, Robert J

AU - Somerset, Douglas W B

PY - 2007/1/1

Y1 - 2007/1/1

N2 - 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.

AB - 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.

KW - C-algebra

KW - inner derivation

KW - primitive ideal

KW - primal ideal

KW - spectral theory

KW - geometric optimisation

KW - inner derivations

KW - primal ideals

KW - norm

U2 - 10.1016/j.jfa.2006.06.012

DO - 10.1016/j.jfa.2006.06.012

M3 - Article

VL - 242

SP - 247

EP - 271

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -