Given a von Neumann algebra $M$, let
$$
 S(M) = \{u\in M: uu^\*u=u\}
 $$
be the set of partial isometries in $M$. Given $u,v\in S(M)$, it is well known that $uv \in S(M)$, provided $u^\*u$ commutes
with $vv^\*$.


Now suppose that $N$ is another von Neumann algebra and that
$$
 \varphi :S(M)\to S(N)
 $$
is a bijective function such that:


1. for every $u,v\in S(M)$, such that $u^\*u$ commutes with $vv^\*$, one has that
$\varphi (u)^\*\varphi (u)$ commutes with $\varphi (v)\varphi (v^\*)$, and
$\varphi (uv)=\varphi (u)\varphi (v)$.
2. $\varphi ^{-1}$ satisfies (1).


Does it follow that $\varphi $ extends to an isomorphism from $M$ to $N$?



Von Neumann algebras with isomorphic sets of partial isometries
No, consider the map $\phi: u \mapsto u^\*$ from the partial isometries in $M$ to the partial isometries in its opposite algebra $M^{op}$. One easily checks that it has the desired properties, but it cannot extend to a linear map because already e.g. $\phi(iI) = -iI \neq iI = i\phi(I)$.


If $M$ is not isomorphic to its opposite algebra, then we have an example where one has a map of the type described between non-isomorphic von Neumann algebras.


