Isometries of tridiagonal algebras (Q582552)

Alg \({\mathcal L}\) denotes the algebra of bounded operators which leave invariant all of the elements in \({\mathcal L}\) that is a family of subspaces. When Alg \({\mathcal L}\) is some tridiagonal algebra, the author proves that if \(\phi: Alg {\mathcal L}\to Alg {\mathcal L}\) is a linear surjective isometry, then there exist unitary operators W and V such that \(\phi (A)=WAV\) for all \(A\in Alg {\mathcal L}.\) This algebras have been found to be useful counterexamples to a number of plausible conjectures, for example, this has non-trivial cohomology.