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Spatiality of isomorphisms between certain reflexive algebras

Authors: M. S. Lambrou and W. E. Longstaff
Journal: Proc. Amer. Math. Soc. 122 (1994), 1065-1073
MSC: Primary 47D25
MathSciNet review: 1216818
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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 $.

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Article copyright: © Copyright 1994 American Mathematical Society

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