Monotone complete C^*-algebras and their automorphism groups

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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.