Conservative derivations and dissipative Laplacians
In this short note the authors prove a noncommutative extension of a theorem stated earlier by \textit{C. J. K. Batty}, the first author and \textit{D. W. Robinson} [The heat semigroup, derivations and Reynolds identity (preprint, 1987)]. They consider a (noncommutative) \(C^*\)-algebra A and a finite sequence of (not necessarily closed) *-derivations \(\delta_ 1,...,\delta_ d\), such that the Laplacian \(\Delta =- \sum^{d}_{k=1}\delta^ 2_ k\) is densely defined on the set \(D(\Delta)=\cap^{d}_{k=1}D(\delta^ 2_ k)\). Under these assumptions the authors prove the equivalence of the following conditions: i) The Laplacian \(\Delta\) is a dissipative operator. ii) For every \(k\in \{1,...,d\}\) the derivation \(\delta_ k\) is a conservative operator on D(\(\Delta)\). A counterexample given by the authors shows that the implication i)\(\Rightarrow ii)\) is not valid for general \(B^*\)-algebra, while the implication ii)\(\Rightarrow i)\) is of course valid in a general Banach space setting. The authors use this theorem to investigate the integrability of finite- dimensional Lie algebras of closed derivations acting on A. Let g be a finite-dimensional Lie algebra of closed derivations acting on A. Let \(A^{\infty}\) be all \(C^{\infty}\)-elements for g, and let \(\delta_ 1,...,\delta_ d\) be a basis of g: Then g exponentiates to a representation of the universal covering group G of the Lie algebra g as a group of *-automorphisms of A if and only if the following three conditions holds: i) The domain D(\(\Delta)\) of the Laplacian is a joint core for the derivations \(\delta_ 1,...,\delta_ d.\) ii) The closure \({\bar \Delta}\) of the Laplacian \(\Delta\) exists, and the closure is a generator of a contraction semigroup \(\{S_ t\}_{t>0}\), such that \(S_ tA\subset A^{\infty}\) \((t>0).\) iii) There exists a constant \(C>0\) such that \(\| \delta_ kS_ t\| \leq Ct^{-1/2}\) for \(0<t\leq 1\), and \(k\in \{1,...,d\}\).
integrability of finite-dimensional Lie algebras of closed derivations
scientific article; zbMATH DE number 4124395