On invariant subalgebras of group C* and von Neumann algebras

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 $\backslash G$ is \emph{just-infinite}, namely every non-trivial normal subgroup of $\backslash G$ has finite index in $\backslash G$.

In Chapter 1, we present some basic aspects of Operator Algebras. In particular, we give a brief introduction on $C\ast$-algebras and the theory of unitary representations of discrete groups. Then, we present some fundamental properties of unitary representations and their relation with the theory of $C\ast$- and $W\ast$-dynamical systems.

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 $C\ast$- and $W\ast$-dynamical systems. We close this chapter via quoting a Factor Theorem that was proved recently in \cite{BoutHoud}. This work plays a central role in the aforementioned developments.

In Chapter 3, we introduce first the concept of co-finiteness for $\backslash G$-invariant subalgebras of $W\pi \ast (\backslash G)$, for a fixed unitary representation $\pi $ of a discrete group $\backslash G$. Then, we introduce operator algebraic analogues of just-infiniteness and show that certain higher rank lattices satisfy them. Moreover, we prove that these properties are strongly dependent on the representation and preserve various operator algebraic properties, such as injectivity and Haagerup property. Finally, for these lattices $\backslash G$ we give a complete description of all $\backslash G$-invariant von Neumann subalgebras of the group von Neumann algebra $\backslash cL(\backslash G)$. In particular, we show that there is a one to one correspondence between the set of normal subgroups of $\backslash G$ and the set of all $\backslash G$-invariant von Neumann subalgebras of $\backslash cL(\backslash G)$.