Automorphisms of operator algebras and quantum measures

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We showed that when M is a von Neumann algebra with a non atomic center then an easy argument can be given to establish the boundedness of completely additive quantum measures ou M.In particular, if {n_j} is a sequence of positive integers and, for each j, A_j ia an abelian von Neumann algebra. with no minimal projections, and M_j is the algebra of n_j by n_j matrices then Σ_j【symmetry】(M_j 【cross product】A_j) is a von Neumann algebra not covered by the Dorofeev-Shertsnev theorem but one to which our results apply. By combining the results obtained here with their deep theorems (specialized to factors) the best possible result is obtained.Let M be a von Neumann algebra which does not have any direct summand isomorphic to the algebra of n by n matrices (for n an integer greater than 1). Then every completely additive quantum measure on M is bounded.2. Let B be any monotone complete C^*-algebra and let G be any locally compact separable Hausdorff group. We gave necessary and sufficient c … More onditions on the (B, G) for the existence of an action α of G on B as a group of *-automorphisms in such a way that (B, G, α) is an admissible dynamical system. Roughly, it is a monotone complete C^*-dynamical system (B, G, α) for which we can construct a monotone complete cross-product B x_α G with the canonical embedding of B.Furthermore, when G is abelian, we can define a dual action of G in such a way that the duality principle of Takesaki is valid.3. We constructed non-trivial examples of admissible monotone complete C^*-dynamic systems. In particular, we constructed such a system where G is the additive group R of real numbers or the Torus T, and where B is the Generic Dynamics Factor A.4. Let Out(A) = Aut(A)/Inn(A) be the outer automorphism group of A.Then, for each integer p with p 【greater than or equal】 2 and each complex number γ with γ^p=1, we constructed periodic automorphisms of A with Connes' outer conjugacy invariant (p, γ) of outer periodicity.5. For any countable discrete group G, it is shown that G can be isomorphically embedded in Out(A). Less