On partial isometries

Question

To study the projection theory in von Neumann algebras we need to study partial isometries, now my question what is analogue of partial isometries in abelian von Neumann algebra namely $L^\infty$?? One more thing is how Murray von Neumann equivalence classifies the set of projections, means like the infinite dimensional case it identifies each dimensional projections, what about infinite case??

Answer 1

In an abelian algebra, a partial isometry satisfies $v^*v=vv^*$, so no two distinct projections are equivalent. So, in $L^\infty$, the partial isometries are those $f$ such that $|f|^2=1_E$ for some measurable set $E$. As the right-hand side only takes the values $0$ and $1$, we can re-write the inequality as $$|f|=1_E.$$

As for the equivalence of projections in the infinite case, the interpretation will depend on the algebra. In $B(H)$, for example, the situation is the same as above: two projections are equivalent if and only if their ranges have the same dimension. But in general you cannot discuss in those terms. The right point of view is to see equivalence as an abstract notion of "same dimension". For instance in a II$_1$ factor you have a trace, that gives you a continuous of "dimensions" of projections; but if you look at the factor represented in $B(H)$, all projections have infinite-dimensional rank, and you cannot distinguish them in $B(H)$.