Classify the von Neumann factors [math]A\subset B(H)[/math] by using as invariant the ordered semigroup formed by the equivalence classes of projections [math]p\in A[/math].

Find a direct proof for the fact that the traces of projections in [math]L(\Gamma)[/math] can take any values in [math][0,1][/math], for an ICC group [math]\Gamma[/math] of your choice.

Do something, statement and proof, even modest, in relation with the von Neumann algebras [math]L(\Gamma)=L^\infty(G)[/math] of the discrete quantum groups [math]\Gamma=\widehat{G}[/math].

Fully clarify the basic properties of the [math]{\rm II}_\infty[/math] factors, and the related construction of the coupling constant.

Prove that we have the formula

for any inclusion of ICC groups [math]\Lambda\subset\Gamma[/math]