Research Projects

The motivation of this project is to classify a class of von Neumann algebras which are coming as a crossed product of abelian von Neumann algebra and a second countable locally compact group. Every probability measure corresponds an abelian von Neumann algebra. An unital self adjoint subalgebra of the algebra of bounded linear opeartors on a separable Hilbert space is called von Neumann algebra if it is closed with respect to the weak operator topology. A factor is a von Neumann algebra if its algebraic center is one dimensional. Factors are building blocks for a von Neumann algebra. There are three types factors : type I, type II and type III. Action of a group on a von Neumann algebra is free if the only group element that acts as the identity automorphism on the von Neumann algebra is the identity element of the group and the action is called ergodic if the only elements of the von Neumann algebra that are invariant under the action of the entire group are scalar multiples of the identity element of the von Neumann algebra. Given a von Neumann algebra with an action of a second countable locally compact group one can produce a new von Neumann algebra called crossed product. It contains the starting von Neumann algebra as a subalgbera and also a copy of the group. In particular, if the group is countable discrete group and the action is free then the crossed product is a factor. Consider the right random walk on a second countable locally compact group and consider the Poisson boundary associated with the random walk. It is kown that the action of the group on the abelian von Neumann algebra correspoding to the Poisson boundary is ergodic. Also, we started with a probabilty measure on the group, called starting measure. In this project, we are interested to get a class of groups whose action is free. This is crucial because if the action is free, the crossed product becomes a factor. If the crossed product is a factor, the project questions the classification and the type of the factor, particularly whether the starting measure influences the type. Further, the project investigates how the Poisson boundary can be realized within the framework of von Neumann algebras, with specific attention to the properties of the induced action and the resulting crossed products.