Characterizations of Lie derivations on Kadison–Singer algebras

Abstract

Kadison–Singer algebra (KS-algebra) is a new class of non-self-adjoint operator algebras. In this paper, we mainly study the standardization of Lie derivations on some KS-algebras. In Sect. 2, we prove that if ${\mathcal {L}}$ is a non-trivial KS-lattice in $M_{3}(\mathbb {C})$, then every Lie derivation from $\textrm{Alg}{{\mathcal {L}}}$ into $M_{3}(\mathbb {C})$ is standard. In Sect. 3, we suppose that ${\mathcal {H}}$ is a separable infinite-dimensional Hilbert space, ${\mathcal {N}}$ is a non-trivial nest on ${\mathcal {H}}$ and $\xi$ is a separating vector for ${\mathcal {N}}''$. Let ${\mathcal {L}}$ be a KS-lattice generated by ${\mathcal {N}}$ and the rank-one projection $P_{\xi }$, we prove that every Lie derivation from $\textrm{Alg}{{\mathcal {L}}}$ into $B({{\mathcal {H}}})$ is standard.