Spatiality of isomorphisms between certain reflexive algebras

Abstract:

Two subspaces M and N of a Hilbert space H are in generalized generic position if $M \cap N = {M^ \bot } \cap {N^ \bot } = (0)$ and $\dim ({M^ \bot } \cap N) = \dim (M \cap {N^ \bot })$. If H is separable and both the pairs $\{ {M_1},{N_1}\}$ and $\{ {M_2},{N_2}\}$ are in generalized generic position, then every algebraic isomorphism $\varphi :{\operatorname {Alg}}\{ {M_1},{N_1}\} \to {\operatorname {Alg}}\{ {M_2},{N_2}\}$ is spatially induced, that is, there exists an invertible operator ${T_0} \in \mathcal {B}(H)$ such that $\varphi (B) = {T_0}BT_0^{ - 1}$, for every $B \in {\operatorname {Alg}}\{ {M_1},{N_1}\}$. The proof of this uses the following result: If H is separable, $\mathcal {M} \subseteq H$ is a proper operator range in H, and the operator $T \in \mathcal {B}(H)$ has the property that, for every $W \in \mathcal {B}(H)$ leaving $\mathcal {M}$ invariant, the range of $WT - TW$ is included in $\mathcal {M}$, then the range of $T - \lambda$ is included in $\mathcal {M}$, for some unique scalar $\lambda$.