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。令f为fermion c^* - 代数，让b（f）为其borel^* - 信封。然后存在一个σ理想，m，微薄的理想，使得b（f）/m是f的常规完成f ^^^。此外，存在一个σ理想的n，因此可以用一般的动力学因子来识别b（f）/n。这些是没有正常状态的单调完整III型因子。 M = n可能是合理的。我们表明，通过利用F.2的傅立叶 *自动形态性，这是错误的。我们在遥远的不交流性概括中对经典的brooks-jewett定理进行了完全解决的概括，以实现矢量测量，以涉及均匀的附加序列的均匀连续性，以强限制了强度界限，最终添加了Vector vector测量值。首先，给出了dieudonne定理的非交通延伸，以推论表明每个单调σ完全C^* - 代数是一个粒状空间。在我们的论点过程中，我们发现了两个非交互性绝对连续性的概念，一个是弱的，但另一个足够强，可以成为一个好工具。注意，矢量度量的自然非交通概括是从c^* - 代数到班纳克空间的弱紧凑型操作员，最后，我们以下列方式扩展了它们的结果：让B成为单位单位σ完全C^* - 单位单位c^* - 代数，x Banach空间，banach空间和{t_n} e {t_n} equectors of b by by by b bes x x x。 lim_ <n→∞> t^<**> _ np存在于B中的每个投影P的X norm中，然后{T^<**> _ n}是B^<**>上的点标准收敛。通过使用此功能，我们以以下形式可以完整地解决问题：假设在B上有一个状态μ，使每个T_N相对于μ绝对是绝对连续的。然后，相对于μ，{t_n}绝对是绝对连续的。