### Abstract

Let B be a unital monotone sigma-complete C*-algebra, X a Banach space and (T-n) (n=1, 2,...) a sequence of weakly compact operators from B to X. Our key theorem implies that if limT(n)p exists for each projection p in B, then there exists a weakly compact operator T such that parallel to T ''(n)(b)-T ''(b)parallel to -> 0, for each b in B ''. (An immediate corollary of this is that B is a Grothendieck space.) Let mu be any state of B such that each T-n is strongly absolutely continuous with respect to mu. Then, as a consequence of our key theorem, we obtain the following uniform absolute continuity property. For each epsilon > 0, there exists delta > 0, such that, whenever b is in the unit ball of B and mu(bb*+b*b)<delta, then parallel to T(n)b parallel to <=epsilon for all n. This last result is a far reaching non-commutative generalization of the classical Brooks-Jewett theorem for vector measures.

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

Pages (from-to) | 301-310 |

Number of pages | 10 |

Journal | Quarterly Journal of Mathematics |

Volume | 56 |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- SPACE

### Cite this

*Quarterly Journal of Mathematics*,

*56*, 301-310. https://doi.org/10.1093/qmath/hah037

**Operators on sigma-complete C*-algebras.** / Brooks, J K ; Saito, K ; Wright, J D M .

Research output: Contribution to journal › Article

*Quarterly Journal of Mathematics*, vol. 56, pp. 301-310. https://doi.org/10.1093/qmath/hah037

}

TY - JOUR

T1 - Operators on sigma-complete C*-algebras

AU - Brooks, J K

AU - Saito, K

AU - Wright, J D M

PY - 2005

Y1 - 2005

N2 - Let B be a unital monotone sigma-complete C*-algebra, X a Banach space and (T-n) (n=1, 2,...) a sequence of weakly compact operators from B to X. Our key theorem implies that if limT(n)p exists for each projection p in B, then there exists a weakly compact operator T such that parallel to T ''(n)(b)-T ''(b)parallel to -> 0, for each b in B ''. (An immediate corollary of this is that B is a Grothendieck space.) Let mu be any state of B such that each T-n is strongly absolutely continuous with respect to mu. Then, as a consequence of our key theorem, we obtain the following uniform absolute continuity property. For each epsilon > 0, there exists delta > 0, such that, whenever b is in the unit ball of B and mu(bb*+b*b)

AB - Let B be a unital monotone sigma-complete C*-algebra, X a Banach space and (T-n) (n=1, 2,...) a sequence of weakly compact operators from B to X. Our key theorem implies that if limT(n)p exists for each projection p in B, then there exists a weakly compact operator T such that parallel to T ''(n)(b)-T ''(b)parallel to -> 0, for each b in B ''. (An immediate corollary of this is that B is a Grothendieck space.) Let mu be any state of B such that each T-n is strongly absolutely continuous with respect to mu. Then, as a consequence of our key theorem, we obtain the following uniform absolute continuity property. For each epsilon > 0, there exists delta > 0, such that, whenever b is in the unit ball of B and mu(bb*+b*b)

KW - SPACE

U2 - 10.1093/qmath/hah037

DO - 10.1093/qmath/hah037

M3 - Article

VL - 56

SP - 301

EP - 310

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

ER -