Totally symmetric algebras and the similarity problem
Let B(H) be the set of all bounded operators on a Hilbert space H, \({\mathcal F}\) the set of all closed operators F with dense domain. Further, let \({\mathcal A}\) be a closed subalgebra of B(H), \(\delta\) be a derivation (i.e. \(\delta (AB)=A\delta (B)+B\delta (A)\), \(A,B\in D(\delta)\), where D(\(\delta)\) is a dense subsalgebra of \({\mathcal A})\). \(\delta^*\), defined by \(\delta^*(A^*)=\delta (A)\), is a derivation from \({\mathcal A}^*\) into B(H). Let \({\mathcal A}_{\delta}\) be the algebra of operators \[ \left( \begin{matrix} A\\ 0\end{matrix} \begin{matrix} \delta (A)\\ A\end{matrix} \right)\in B(H\oplus H). \] \(\delta\) is closed iff \({\mathcal A}_{\delta}\) is a Banach algebra. An involution {\#} is introduced such that \({\mathcal A}_{\delta^*}=({\mathcal A}_{\delta})^*\). \({\mathcal A}_{\delta}\) is totally symmetric if \(\delta\) is a closed *-involution. For \(\delta_ F(A):=i[F,A]\) (F selfadjoint), \({\mathcal A}_{\delta_ F}\) are reflexive, semi-simple, totally symmetric algebras. For unbounded F, \({\mathcal A}_{\delta_ F}\) has a strictly irreducible representation not being similar to *-representations.
reflexive, semi-simple, totally symmetric algebras
strictly irreducible representation not being similar to *-representations
scientific article; zbMATH DE number 4094214
scientific article