A class of closed manifolds in nonseparable Hilbert spaces

In this paper, we consider a class of generalized closed linear manifolds in a nonseparable Hilbert space H, which is closely related to the generalized Fredholm theory. We first investigate properties of the set \({\mathcal {B}}_{\vartriangleleft }=\{T\in {\mathcal {M}}:\overline{T(H)}\subset A(H)\) for some \(A\in {\mathcal {B}}\},\) where \({\mathcal {B}}\) is a \(C^*\)-subalgebra of a von Neumann algebra \({\mathcal {M}}\). It is proved that a selfadjoint \({\mathcal {B}}_{\vartriangleleft }\) is always an ideal in \({\mathcal {M}}\). In a type \(\textrm{II}_\infty \) factor, we show that there exists a tracial weight (whose range containing infinite cardinals) such that two projections are equivalent if and only if they have the same tracial weight, which leads to a complete characterization of such selfadjoint \({\mathcal {B}}_{\vartriangleleft }\) when \({\mathcal {M}}\) is a factor. Then we introduce the concept of closed manifolds with respect to a pair of C*-algebras and study some properties. Finally, when m is an infinite cardinal, as a special important case we focus on m-closed subspaces and operators which preserve m-closed subspaces. It is proved that these operators are either of rank less than m, or the generalized left semi-Fredholm operators.