What is the most general statement we can make about the Hilbert spaces with isomorphic endomorphism algebras? When are we allowed to conclude that the Hilbert spaces themselves are isomorphic?
The specific case I have in mind is when two Hilbert spaces have isomorphic C*-algebras of operators.
From $B(H)$, you can recover the dimension of $H$ as the maximal cardinality of pairwise orthogonal families of projections. So $$ B(H)\simeq B(K)\ \text{ (as C$^*$-algebras) } \iff H\simeq K. $$