### Abstract

Let A be a C*-algebra and let K be a relatively weakly compact subset of the dual of A. Let psi be a positive linear functional on A such that, for each phi in K, phi is strongly absolutely continuous with respect to psi. Then, for each epsilon > 0, there exists delta > 0, such that for each x in the closed unit ball of A, psi (xx* + x*x) 1/2_less than or equal to delta implies \phi(x)\ less than or equal to epsilon for every phi is an element of K. This result is extended to the situation where K is a sigma-bounded set of weakly compact operators from A to a Banach space Y.

Original language | English |
---|---|

Pages (from-to) | 31-40 |

Number of pages | 9 |

Journal | Quarterly Journal of Mathematics |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 |

### Cite this

*C**-algebras is automatically uniform.

*Quarterly Journal of Mathematics*,

*55*(1), 31-40. https://doi.org/10.1093/qmath/hag037

**When absolute continuity on C*-algebras is automatically uniform.** / Brooks, J. K.; Saito, Kazuyuki; Wright, John David Maitland.

Research output: Contribution to journal › Article

*C**-algebras is automatically uniform',

*Quarterly Journal of Mathematics*, vol. 55, no. 1, pp. 31-40. https://doi.org/10.1093/qmath/hag037

*C**-algebras is automatically uniform. Quarterly Journal of Mathematics. 2004;55(1):31-40. https://doi.org/10.1093/qmath/hag037

}

TY - JOUR

T1 - When absolute continuity on C*-algebras is automatically uniform

AU - Brooks, J. K.

AU - Saito, Kazuyuki

AU - Wright, John David Maitland

PY - 2004

Y1 - 2004

N2 - Let A be a C*-algebra and let K be a relatively weakly compact subset of the dual of A. Let psi be a positive linear functional on A such that, for each phi in K, phi is strongly absolutely continuous with respect to psi. Then, for each epsilon > 0, there exists delta > 0, such that for each x in the closed unit ball of A, psi (xx* + x*x) 1/2_less than or equal to delta implies \phi(x)\ less than or equal to epsilon for every phi is an element of K. This result is extended to the situation where K is a sigma-bounded set of weakly compact operators from A to a Banach space Y.

AB - Let A be a C*-algebra and let K be a relatively weakly compact subset of the dual of A. Let psi be a positive linear functional on A such that, for each phi in K, phi is strongly absolutely continuous with respect to psi. Then, for each epsilon > 0, there exists delta > 0, such that for each x in the closed unit ball of A, psi (xx* + x*x) 1/2_less than or equal to delta implies \phi(x)\ less than or equal to epsilon for every phi is an element of K. This result is extended to the situation where K is a sigma-bounded set of weakly compact operators from A to a Banach space Y.

U2 - 10.1093/qmath/hag037

DO - 10.1093/qmath/hag037

M3 - Article

VL - 55

SP - 31

EP - 40

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

IS - 1

ER -