We study asymptotic behavior of a Markov semigroup on a von-Neumann algebra by exploring a maximal von-Neumann subalgebra where the Markov semigroup is an automorphism. This enables us to prove that strong mixing is equivalent to ergodic property for continuous time Markov semigroup on a type-I von-Neumann algebra with center completely atomic. For discrete time dynamics we prove that an aperio...

Ulam [1] gave a talk before theMathematics Club of the University ofWisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. We are given a group G and a metric group G with metric ρ( , ). Given > 0, does there exist a δ > 0 such that if f :G G satisfies ρ( f (xy), f (x) f (y)) < δ for all x, y G, then a hom...

The term von Neumann computer has two common meanings. Its strictest de nition refers to a speci c type of computer organization, or \architecture," in which instructions and data are stored together in a common memory. This type of architecture is distinguished from the \Harvard" architecture in which separate memories are used to store instructions and data. The term \von Neumann computer" al...

1. Topologies on B(H) and the double commutant theorem 2 2. Examples of von Neumann algebras 5 2.1. Abelian von Neumann algebras 6 2.2. Group von Neumann algebras and representation theory 6 2.3. The Hyperfinite II1 factor R 9 3. Projections, Type Classification, and Traces 10 3.1. Projections and the spectral theorem 10 3.2. Type classification of factors 12 3.3. Traces 13 4. Tracial ultrapowe...

In this paper, we clarify the structure of the Stone spectrum of an arbitrary finite von Neumann algebra R of type In. The main tool for this investigation is a generalized notion of rank for projections in von Neumann algebras of this type.

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We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any non-injective subfactor in a hyperbolic group von Neumann algebra is non-Γ and prime. The proof is based on C∗-algebra theory.

We prove that the von Neumann algebras generated by n q-Gaussian elements, are factors for n > 2.