All-derivable points in nest algebras

Details All-derivable points in nest algebras All-derivable points in nest algebras

2010-08-09

arxiv.org/abs/1008.1434v3

Linear Algebra and Its Applications 2010

Mathematics

Operator Algebras

12 pages, latex

Scientific paper

10.1016/j.laa.2010.01.034

Suppose that $\mathscr{A}$ is an operator algebra on a Hilbert space $H$. An element $V$ in $\mathscr{A}$ is called an all-derivable point of $\mathscr{A}$ for the strong operator topology if every strong operator topology continuous derivable mapping $\phi$ at $V$ is a derivation. Let $\mathscr{N}$ be a complete nest on a complex and separable Hilbert space $H$. Suppose that $M$ belongs to $\mathscr{N}$ with $\{0\}\neq M\neq\ H$ and write $\hat{M}$ for $M$ or $M^{\bot}$. Our main result is: for any $\Omega\in alg\mathscr{N}$ with $\Omega=P(\hat{M})\Omega P(\hat{M})$, if $\Omega |_{\hat{M}}$ is invertible in $alg\mathscr{N}_{\hat{M}}$, then $\Omega$ is an all-derivable point in $alg\mathscr{N}$ for the strong operator topology.

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