# On invariant subalgebras of group C* and von Neumann algebras

## Abstract

This thesis focuses on the structure theory of simple C* and von Neumann algebras generated by unitary representations of irreducible lattices in higher rank semisimple Lie groups. The seminal work done by Furstenberg and Margulis, inspired recently a whole body of mathematical work in Operator Algebra Theory. \cite{Bek07},\cite{BoutHoud}, \cite{BKKO}, \cite{KalKen}, \cite{Pet15}.
In this thesis, we introduce operator algebraic properties which are analogous to the group theoretic property of just-infiniteness and show that certain higher rank lattices satisfy them. Already,

Margulis' Normal Subgroup Theorem (NST) \cite{Marg-book} states that, every higher rank lattice

In Chapter 1, we present some basic aspects of Operator Algebras. In particular, we give a brief introduction on

In Chapter 2, we give a historical overview of various generalizations of Margulis' NST in the Operator Algebra framework, as well as their connection with Margulis' Superrigidity Theorem and the theory of

In Chapter 3, we introduce first the concept of co-finiteness for