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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是费米子C^*-代数，B（F）是它的Borel^*-包络.则存在一个σ-理想M，即贫乏理想，使得B（F）/M是F的正则完备化F^^^.另外，存在一个σ-理想N，使得B（F）/N可以用一般动力学因子来表示.这些因子是单调完全的III型因子，没有正规态。这可能是合理的，M=N。利用F.2的Fourier *-自同构证明了这是错误的.我们完全解决了经典向量测度Brooks-Jewett定理的一个非交换推广问题，该推广涉及σ-域上强有界的可加向量测度序列的一致绝对连续性.首先，给出了Dieudonne定理的一个非交换推广及其推论，即每个单调σ-完备C^*-代数都是Grothendieck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的概念，一个是弱的，但另一个是强到足以成为一个很好的工具。注意到向量测度的自然非交换推广是从C^*-代数到Banach空间的弱紧算子，最后，我们推广了他们的结果：设B是单位单调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.如果对于B中的每个投影p，lim_<n→∞>T^<**>_np存在于X的范数中，则{T^<**>_n}在B^<**>上是点范数收敛的.利用这一点，我们得到了如下形式的问题的完全解：假设存在B上的状态μ，使得每个T_n关于μ是强绝对连续的。则{T_n}关于μ一致强绝对连续。