Monotone complete C^*-algebras and their automorphism groups

单调完备C^*-代数及其自同构群

基本信息

批准号：
14540197
负责人：
SAITO Kazuyuki
金额：
$ 0.96万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

1.Let F be the Fermion C^*-algebra and let B(F) be its Borel^*-envelope. Then there exists a σ-ideal, M, the meagre ideal, such that B(F)/M is the regular completion F^^^of F. Also, there exists a σ-ideal N such that B(F)/N can be identified with the generic dynamics factor. These are monotone complete type III factor which have no normal states. It might be plausible that, M=N. We show that this is false by making use of the Fourier *-automorphism of F.2.We have a complete solution of a problem on a far reaching non-commutative generalization of the classical Brooks-Jewett theorem for vector measures concerning the uniform abosolute continuity of a sequence of strongly bounded, finitely additve vector measures on a σ-field. First of all, a non-commutative extension of Dieudonne's theorem is given with a Corollary which state that each monotone σ-complete C^*-algebra is a Grothendieck space. In the course of our arguments, we found two notion of non-commutative absolute continuity, one is weak but the other is strong enough to be a good tool. Noting that the natural non-commutative generalization of a vector measure is a weakly compact operator from a C^*-algebra to a Banach space, finally, we extend their result in the following way : Let B be a unital monotone σ-complete C^*-algebra, X a Banach space and {T_n} a sequence of weakly compact operators from B to X. If lim_<n→∞>T^<**>_np exists in the norm of X for each projection p in B, then {T^<**>_n} is point norm convergent on B^<**>. By using this, we have a complete solution of the problem in the following form : Suppose that there is a state μ on B such that each T_n, is strongly absolutely continuous with respect to μ. Then {T_n} is uniformly strongly absolutely continuous with respect to μ.

1.设 F 为费米子 C^* 代数，B(F) 为其 Borel^* 包络，则存在 σ 理想 M，即贫理想，使得 B(F)/M 为F 的正则完成 F^^^。此外，存在一个 σ-理想 N，使得 B(F)/N 可以用通用动态因子来识别。这些是单调完整的 III 型因子，可能没有正常状态。看似合理我们通过利用 F.2 的傅里叶 *-自同构证明这是错误的。我们对经典 Brooks-Jewett 定理的深远非交换推广问题有了完整的解决方案关于 σ 域上的强有界、有限可加向量测度序列的一致绝对连续性的向量测度首先，给出了 Dieudonne 定理的非交换扩展。推论表明每个单调 σ-完备 C^*-代数都是一个格洛腾迪克空间在我们的论证过程中，我们发现了两个非交换绝对连续性的概念，一个是弱的，但另一个是足够强的。一个很好的工具，注意到向量测度的自然非交换泛化是从 C^*-代数到 Banach 空间的弱紧算子，最后，我们通过以下方式扩展它们的结果：令 B 为 a单位单调 σ-完备 C^*-代数，X 是 Banach 空间，{T_n} 是从 B 到 X 的弱紧算子序列。如果 lim_<n→∞>T^<**>_np 存在于 X 的范数中对于 B 中的每个投影 p，则 {T^<**>_n} 是收敛于 B^<**> 的点范数。通过使用它，我们可以得到以下形式的问题的完整解：假设有一个B 上的状态 μ 使得每个 T_n 对于 μ 是强绝对连续的，则 {T_n} 对于 μ 是一致强绝对连续的。