Isomorphisms of some reflexive algebras
Volume 187 / 2008
Studia Mathematica 187 (2008), 95-100
MSC: Primary 47B47, 47L35.
DOI: 10.4064/sm187-1-5
Abstract
Suppose $\mathcal L_1$ and $\mathcal L_2$ are subspace lattices on complex separable Banach spaces $X$ and $Y$, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from $\mathop{\rm alg}\mathcal L_1$ to $\mathop{\rm alg}\mathcal L_2$ is quasi-spatial; in particular, if a subspace lattice $\mathcal L$ of a complex separable Banach space $X$ contains a sequence $E_i$ such that $(E_i)_ -\neq X$, $E_i \subseteq E_{i+1}$, and $\bigvee_{i=1}^{\infty} E_i = X$ then every automorphism of $\mathop{\rm alg} \mathcal L$ is quasi-spatial.
