On derivations annihilating a maximal abelian subalgebra

Abstract:

Let $\mathcal {A}$ be an ${\text {AF}}\;{C^\ast }$-algebra, and let $\delta$ be a closed $\ast$-derivation which annihilates the maximal abelian subalgebra $\mathcal {C}$ of diagonal elements of $\mathcal {A}$. Then we show that $\delta$ generates an approximately inner ${C^\ast }$-dynamics on $\mathcal {A}$, and that $\delta$ is a commutative $\ast$-derivation. Any two closed $\ast$-derivations vanishing on $\mathcal {C}$ are shown to be strongly commuting. More generally, if $\delta$ is a semiderivation on $\mathcal {A}$ which vanishes on $\mathcal {C}$, we prove that $\delta$ is a generator of a semigroup of strongly positive contractions of $\mathcal {A}$.