We approach the study of sub-von Neumann algebras of the group von Neumann algebra L(\Gamma) for a countable group \Gamma from a dynamical perspective. We show that L(\Gamma) admits a maximal \Gamma-invariant amenable subalgebra. We introduce the notion of Invariant Random Algebra (IRA) in the space of sub-algebras, analogous to the concept of Invariant Random Subgroup (IRS). Finally, we shall show that amenable IRAs are supported on the maximal amenable \Gamma-invariant subalgebra of L(\Gamma). 

 

This is joint work with Yair Hartman and Hanna Oppelmayer. 

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